5-probability-theory-tutorial.ipynb

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   "source": [
    "## Google Colab Rust Setup\n",
    "\n",
    "The following cell is used to set up a Rust environment on Colab. Don't execute it locally!\n",
    "\n",
    "Many thanks to [`mateusvmv`](https://github.com/mateusvmv) for this hack in [`gist.github.com/korakot/ae95315ea6a3a3b33ee26203998a59a3`](https://gist.github.com/korakot/ae95315ea6a3a3b33ee26203998a59a3?permalink_comment_id=4715636#gistcomment-4715636)."
   ]
  },
  {
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   "id": "1d278011-f31a-4e13-890f-1ad8533396b8",
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   "source": [
    "# This script sets up and spins up a Jupyter Notebook environment with a Rust kernel using Nix and IPC Proxy. \n",
    "!wget -qO- https://gist.github.com/wiseaidev/2af6bef753d48565d11bcd478728c979/archive/3f6df40db09f3517ade41997b541b81f0976c12e.tar.gz | tar xvz --strip-components=1\n",
    "!bash setup_evcxr_kernel.sh"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ac5c5759",
   "metadata": {},
   "source": [
    "## Install Dependencies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2b125d4c",
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   "source": [
    ":dep statrs = { version = \"0.16.0\" }\n",
    "// or\n",
    "// :dep statrs = { git = \"https://github.com/statrs-dev/statrs\" }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "581695d8-ef17-46f8-a5cd-58eedfa1a1e2",
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   "source": [
    ":dep plotters = { version = \"^0.3.5\", default_features = false, features = [\"evcxr\", \"all_series\"] }\n",
    "// or\n",
    "// :dep statrs = { git = \"https://github.com/plotters-rs/plotters\" }"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7082da10",
   "metadata": {},
   "source": [
    "## Import Libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "f491c1ee",
   "metadata": {},
   "outputs": [],
   "source": [
    "use plotters::prelude::*;\n",
    "use statrs::statistics::Distribution;\n",
    "use statrs::distribution::{Continuous, Discrete};\n",
    "use statrs::distribution::{Normal, Binomial, Geometric, Hypergeometric, Poisson, Exp, Gamma, ChiSquared, Beta, NegativeBinomial, Laplace};"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9372f9d0",
   "metadata": {},
   "source": [
    "# 1. Normal Distribution"
   ]
  },
  {
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   "source": [
    "## Probability Density Function (PDF)\n",
    "\n",
    "### $f(x) = \\frac{1}{{\\sigma \\sqrt{2\\pi}}} e^{-\\frac{1}{2} \\left(\\frac{x - \\mu}{\\sigma}\\right)^2}$\n",
    "\n",
    "Here, x represents the value to be standardized (each individual height), μ represents the mean, and σ represents the standard deviation."
   ]
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   "source": [
    "## Example\n",
    "Heights samples: 168cm, 172cm,165cm ,170cm ,174cm ,180cm, 162cm, etc.\n",
    "\n",
    "## Mean\n",
    "\n",
    "### $\\mu = \\frac{1}{n} \\sum_{i=1}^{n} x_i$\n",
    "\n",
    "### $\\mu = \\frac{\\sum_{i=1}^{n} x_i}{n} $\n",
    "### $\\mu = \\frac{168 + 172 + 165 + 170 + 174 + 180 + 162 + 176 + 168 + 172 + 170 + 178 + 175 + 169 + 171 + 172 + 174 + 167 + 176 + 173}{20}$\n",
    "### $\\mu = \\frac{3432}{20}$\n",
    "### $\\mu = 171.6 cm$\n",
    "\n",
    "## Standard Deviation\n",
    "\n",
    "### $\\sigma = \\sqrt{\\frac{1}{n} \\sum_{i=1}^{n} (x_i - \\mu)^2}$\n",
    "\n",
    "### $\\sigma = \\sqrt{\\frac{\\sum_{i=1}^{n} (x_i - \\mu)^2}{n}}$\n",
    "### $\\sigma = \\sqrt{\\frac{(168 - 171.6)^2 + (172 - 171.6)^2 + \\ldots + (173 - 171.6)^2}{20}}$\n",
    "### $\\sigma = \\sqrt{\\frac{370.8}{20}}$\n",
    "### $\\sigma \\approx \\sqrt{18.54}$\n",
    "### $\\sigma \\approx 4.31cm$\n",
    "\n",
    "## z-score\n",
    "\n",
    "### $z = \\frac{x - \\mu}{\\sigma}$\n",
    "\n",
    "### $z = \\frac{168 - \\mu}{\\sigma}$\n",
    "### $z = \\frac{168 - 171.6}{4.31}$\n",
    "### $z \\approx −0.84$\n",
    "\n",
    "Repeat this calculation for each individual height value to obtain the corresponding z-scores.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb6804c2",
   "metadata": {},
   "source": [
    "## Probability Density Function (PDF)"
   ]
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  {
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       "[(162.0, 0.0077111498014200905), (165.0, 0.0286124002877568), (167.0, 0.05236124932813138), (168.0, 0.0653262210511061), (168.0, 0.0653262210511061), (169.0, 0.07722030750534586), (170.0, 0.08648523542740724), (170.0, 0.08648523542740724), (171.0, 0.09177383324586816), (172.0, 0.09227036241240198), (172.0, 0.09227036241240198), (172.0, 0.09227036241240198), (173.0, 0.08789659179986477), (174.0, 0.07933198103903279), (174.0, 0.07933198103903279), (175.0, 0.06784080894510099), (176.0, 0.05496676666227962), (176.0, 0.05496676666227962), (178.0, 0.03069148492847293), (180.0, 0.01381024648433747)]"
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   "source": [
    "let random_points:Vec<(f64,f64)> = {\n",
    "    let heights_dist = Normal::new(171.6, 4.305).unwrap();\n",
    "    let mut heights = vec![162.0, 165.0, 167.0, 168.0, 168.0, 169.0, 170.0, 170.0, 171.0, 172.0, 172.0, 172.0, 173.0, 174.0, 174.0, 175.0, 176.0, 176.0, 178.0, 180.0];\n",
    "    let y_iter: Vec<f64> = heights.iter().map(|x| heights_dist.pdf(*x)).collect();\n",
    "    heights.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a092f9f6",
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       "<line opacity=\"0.1\" stroke=\"#000000\" stroke-width=\"1\" x1=\"40\" y1=\"35\" x2=\"639\" y2=\"35\"/>\n",
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       "Probability\n",
       "</text>\n",
       "<text x=\"340\" y=\"480\" dy=\"-0.5ex\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "Heights (cm)\n",
       "</text>\n",
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       "<text x=\"30\" y=\"439\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.0\n",
       "</text>\n",
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       "<text x=\"30\" y=\"399\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.01\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,399 39,399 \"/>\n",
       "<text x=\"30\" y=\"359\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.02\n",
       "</text>\n",
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       "<text x=\"30\" y=\"318\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.03\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,318 39,318 \"/>\n",
       "<text x=\"30\" y=\"278\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.04\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,278 39,278 \"/>\n",
       "<text x=\"30\" y=\"237\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.05\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,237 39,237 \"/>\n",
       "<text x=\"30\" y=\"197\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.06\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,197 39,197 \"/>\n",
       "<text x=\"30\" y=\"157\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.07\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,157 39,157 \"/>\n",
       "<text x=\"30\" y=\"116\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.08\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,116 39,116 \"/>\n",
       "<text x=\"30\" y=\"76\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.09\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,76 39,76 \"/>\n",
       "<text x=\"30\" y=\"35\" dy=\"0.5ex\" text-anchor=\"end\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "0.1\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"34,35 39,35 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"40,440 639,440 \"/>\n",
       "<text x=\"40\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "162.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"40,440 40,445 \"/>\n",
       "<text x=\"106\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "164.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"106,440 106,445 \"/>\n",
       "<text x=\"173\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "166.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"173,440 173,445 \"/>\n",
       "<text x=\"239\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "168.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"239,440 239,445 \"/>\n",
       "<text x=\"306\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "170.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"306,440 306,445 \"/>\n",
       "<text x=\"372\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "172.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"372,440 372,445 \"/>\n",
       "<text x=\"439\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "174.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"439,440 439,445 \"/>\n",
       "<text x=\"505\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "176.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"505,440 505,445 \"/>\n",
       "<text x=\"572\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "178.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"572,440 572,445 \"/>\n",
       "<text x=\"639\" y=\"450\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "180.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"639,440 639,445 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#FF0000\" stroke-width=\"1\" points=\"40,408 139,324 206,228 239,176 239,176 272,128 306,90 306,90 339,69 372,67 372,67 372,67 406,84 439,119 439,119 472,165 505,217 505,217 572,316 639,384 \"/>\n",
       "<rect x=\"465\" y=\"222\" width=\"170\" height=\"30\" opacity=\"1\" fill=\"#FFFFFF\" stroke=\"none\"/>\n",
       "<rect x=\"465\" y=\"222\" width=\"170\" height=\"30\" opacity=\"1\" fill=\"none\" stroke=\"#000000\"/>\n",
       "<text x=\"505\" y=\"232\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝜇=171.6, 𝜎=4.305\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#FF0000\" stroke-width=\"1\" points=\"475,237 495,237 \"/>\n",
       "</svg>\n",
       "</div>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Normal Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(162f64..180f64, 0f64..0.1f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Heights (cm)\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"𝜇=171.6, 𝜎=4.305\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30f3d9a0",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0582420d",
   "metadata": {},
   "source": [
    "# 2. Binomial Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "502bdd6b",
   "metadata": {},
   "source": [
    "## Binomial Coefficient\n",
    "\n",
    "### $\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$\n",
    "\n",
    "## Binomial Probability\n",
    "\n",
    "### $P(X=k) = \\binom{n}{k} \\cdot p^k \\cdot q^{n-k}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80b2ec77",
   "metadata": {},
   "source": [
    "## Example\n",
    "- 100 individuals surveyed (**n**) voting for a particular candidate.\n",
    "- The probability of success is denoted by **p** and remains unknown at this point.\n",
    "- **k** refers to how many people voted positively towards the given candidate.\n",
    "- Suppose we analyze voting trends and deduce that there's a 0.6 chance of casting votes in favor of this candidate (p).\n",
    "\n",
    "### Calculate the probability\n",
    "\n",
    "### $P(X=k) = \\binom{n}{k} \\cdot p^k \\cdot q^{n-k}$\n",
    "### $P(X=k) = \\binom{100}{k} \\cdot (0.6)^k \\cdot (0.4)^{100-k}$\n",
    "\n",
    "By adjusting various **k** values, we can accurately calculate the probabilities linked with achieving varying numbers of respondents who have voted for the candidate in this survey."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78bc4961",
   "metadata": {},
   "source": [
    "## Probability Mass Function (PMF)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "33e01c22",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 1.6069380442590007e-40), (1.0, 2.4104070663884447e-38), (2.0, 1.789727246793468e-36), (3.0, 8.769663509287653e-35), (4.0, 3.1899651015033815e-33), (5.0, 9.187099492329999e-32), (6.0, 2.1819361294284227e-30), (7.0, 4.3950427749913184e-29), (8.0, 7.663855838891474e-28), (9.0, 1.1751245619633727e-26)]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Binomial::new(0.6, 100).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 100).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "70a2703e",
   "metadata": {},
   "outputs": [
    {
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       "Binomial Distribution\n",
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Binomial Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..100f64, 0f64..0.1f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Individuals\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"p=0.6, n=100\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c23f50c4",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58643b37",
   "metadata": {},
   "source": [
    "## 3. Geometric Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c39f72a6",
   "metadata": {},
   "source": [
    "## Probability Mass Function (PMF)\n",
    "\n",
    "### $P(X = x) = p \\cdot (1 - p)^{(x-1)}$\n",
    "\n",
    "In this equation, x represents any one of the repeated values of the random variable X, which represents the number of trials needed until the first occurrence of a successful event A. The term (1 - p) represents the probability of an event other than A (i.e., failure), denoted as P(Ā), while p represents the probability of event A (i.e., success), denoted as P(A)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd64db62",
   "metadata": {},
   "source": [
    "## Mean\n",
    "\n",
    "### $E(X) = \\frac{1}{p}$\n",
    "\n",
    "## Variance\n",
    "\n",
    "### $Var(X) = \\frac{1 - p}{p^2}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d190a2ab",
   "metadata": {},
   "source": [
    "## Example\n",
    "\n",
    "Suppose we are flipping a fair coin until we get heads for the first time. What is the probability that it takes exactly 4 flips to achieve the first head? In this case, we have a success probability of p = 0.5 since the coin is fair. We can use the Geometric distribution formula to calculate the probability:\n",
    "\n",
    "### $P(X = 4) = 0.5 \\cdot (1 - 0.5) ^{(4-1)} = 0.0625$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd8b493f",
   "metadata": {},
   "source": [
    "## Probability Mass Function (PMF)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a58bb2fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 0.0), (1.0, 0.2), (2.0, 0.16000000000000003), (3.0, 0.12800000000000003), (4.0, 0.10240000000000003), (5.0, 0.08192000000000005), (6.0, 0.06553600000000004), (7.0, 0.05242880000000003), (8.0, 0.041943040000000036), (9.0, 0.03355443200000003)]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Geometric::new(0.2).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 10).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Geometric::new(0.5).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 10).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Geometric::new(0.8).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 10).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8c80b747",
   "metadata": {},
   "outputs": [
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       "Geometric Distribution\n",
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Geometric Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..8f64, 0f64..0.9f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"p=0.2\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"p=0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"p=0.8\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16e2eaec",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce944211",
   "metadata": {},
   "source": [
    "## 4. Hypergeometric Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5cc778c0",
   "metadata": {},
   "source": [
    "### Probability Mass Function (PMF)\n",
    "\n",
    "### $P(X = x) = \\frac{{\\binom{M}{x} \\cdot \\binom{N-M}{n-x}}}{{\\binom{N}{n}}}$\n",
    "\n",
    "In this equation, X represents the random variable, and x represents the number of observed (sample) successes. Let's consider the follwoing example: Suppose there is a bag containing 100 marbles, out of which 20 marbles are red and 80 marbles are blue. We randomly select 10 marbles from the bag without replacement. We want to calculate the probability of obtaining exactly 3 red marbles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "6d78e1ed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 7.360755374842311e-6), (1.0, 0.00010485406516867851), (2.0, 0.0007225100428030384), (3.0, 0.0032093363940940488), (4.0, 0.010332884619119435), (5.0, 0.02570705266368036), (6.0, 0.05145021073840004), (7.0, 0.0851532899615837), (8.0, 0.11889944363004967), (9.0, 0.1421935778973869)]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Hypergeometric::new(500, 50, 100).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 60).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Hypergeometric::new(500, 60, 200).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 60).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Hypergeometric::new(500, 70, 300).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 60).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "9d3d764c",
   "metadata": {},
   "outputs": [
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     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Hypergeometric Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..60f64, 0f64..0.16f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"N=500, M=50, n=100\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"N=500, M=60, n=200\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"N=500, M=70, n=300\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62caa2ed",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6c76470",
   "metadata": {},
   "source": [
    "## 5. Poisson distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd100e83",
   "metadata": {},
   "source": [
    "### Probability Mass Function (PMF)\n",
    "\n",
    "### $P(x=r) = \\frac{e^{-\\lambda} \\lambda^r}{r!}$\n",
    "\n",
    "In this equation, P(x=r) represents the probability of the event occurring r times, λ represents the average or expected number of occurrences, and e is the base of the natural logarithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "80f7901a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 0.36787944117144233), (1.0, 0.36787944117144233), (2.0, 0.18393972058572114), (3.0, 0.06131324019524039), (4.0, 0.015328310048810101), (5.0, 0.00306566200976202), (6.0, 0.0005109436682936698), (7.0, 7.299195261338139e-5), (8.0, 9.123994076672672e-6), (9.0, 1.013777119630298e-6)]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Poisson::new(1.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 20).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Poisson::new(5.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 20).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Poisson::new(10.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0 .. 20).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pmf(*x as u64)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "795c5564",
   "metadata": {},
   "outputs": [
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       "𝜆=1.0\n",
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       "𝜆=1.0\n",
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       "𝜆=1.0\n",
       "</text>\n",
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     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Poisson Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..20f64, 0f64..0.4f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"𝜆=1.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"𝜆=5.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"𝜆=10.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57ed4034",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ae3bfc3",
   "metadata": {},
   "source": [
    "## 6. Gumbel Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38b95882",
   "metadata": {},
   "source": [
    "### Probability Density Function (PDF)\n",
    "\n",
    "### $f(x) = \\frac{1}{\\beta}e^{-\\left(\\frac{x-\\alpha}{\\beta}+e^{-\\left(\\frac{x-\\alpha}{\\beta}\\right)}\\right)}$\n",
    "\n",
    "In this equation, α represents the location parameter, which determines the horizontal shift or location of the distribution. It is calculated as:\n",
    "\n",
    "### $\\alpha = \\mu - \\beta \\cdot nc$\n",
    "\n",
    "With the Gumbel Distribution, β dictates how far-reaching its spread will be. Meanwhile, μ represents the average value and nc (approximated at 0.5772) is Euler's constant. β is calculated as:\n",
    "\n",
    "### $\\beta = \\frac{\\sigma}{k}$\n",
    "\n",
    "In this equation, the symbol σ denotes the standard deviation while k is a constant that relates to the distribution's shape. By rearranging and squaring both sides of the above equation, we can compute the variance represented by σ².\n",
    "\n",
    "### $\\sigma^2 = \\frac{\\pi^2}{6} \\cdot \\beta^2$\n",
    "\n",
    "The formula that follows provides the cumulative distribution function (CDF) of the Gumbel Distribution.\n",
    "\n",
    "### $F(x) = e^{-e^{-\\left(\\frac{x-\\alpha}{\\beta}\\right)}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d9198ad",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Suppose we have half a century's worth of data available for this city's yearly highest recorded temperatures, and it has been established that its mean is 28°C with a standard deviation measuring 3.5°C. Our aim now is to determine the probability of experiencing a maximum temperature exceeding 35°C in any given year.\n",
    "\n",
    "### $\\alpha = \\mu - \\beta \\cdot \\gamma$\n",
    "\n",
    "### $\\beta = \\frac{k}{\\sigma}$\n",
    "\n",
    "### $\\beta = \\frac{3.5}{\\sigma}$\n",
    "\n",
    "### $\\beta = \\frac{0.5772}{3.5}$\n",
    "\n",
    "### $F(x) = e^{-e^{(-(\\frac{x-\\alpha}{\\beta}))}}$\n",
    "\n",
    "### $F(35) = e^\t{-e^{(-(\\frac{35-28}{3.5 \\cdot 0.5772}))}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "677b041c",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d89881e5",
   "metadata": {},
   "source": [
    "## 7. Exponential Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0e3f89c",
   "metadata": {},
   "source": [
    "### Probability Density Function (PDF)\n",
    "\n",
    "### $f(x) = \\lambda e^{-\\lambda x}$\n",
    "\n",
    "### Cumulative Distribution Function (CDF)\n",
    "\n",
    "### $F(x) = 1 - e^{-\\lambda x}$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "8a74abf7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 0.5), (0.01, 0.49750623959634116), (0.02, 0.49502491687458405), (0.03, 0.4925559698015313), (0.04, 0.4900993366533776), (0.05, 0.4876549560141663), (0.06, 0.4852227667742541), (0.07, 0.48280270812878323), (0.08, 0.4803947195761616), (0.09, 0.47799874091655)]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Exp::new(0.5).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Exp::new(1.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Exp::new(1.5).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f003b887",
   "metadata": {},
   "outputs": [
    {
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       "𝜆=0.5\n",
       "</text>\n",
       "<text x=\"581\" y=\"232\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝜆=1.0\n",
       "</text>\n",
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       "𝜆=1.5\n",
       "</text>\n",
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       "</svg>\n",
       "</div>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Exponential Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..9f64, 0f64..0.4f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"𝜆=0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"𝜆=1.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"𝜆=1.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0809c9e",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ace976d1",
   "metadata": {},
   "source": [
    "## 8. Gamma Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e4bd3f7",
   "metadata": {},
   "source": [
    "### Probability Density Function (PDF)\n",
    "\n",
    "### $f(x; \\alpha) = \\frac{x^{\\alpha-1} e^{-x}}{\\Gamma(\\alpha)}$\n",
    "\n",
    "### Gamma Function\n",
    "\n",
    "### $\\Gamma(\\alpha) = \\int_0^\\infty x^{\\alpha-1} e^{-x} \\, dx$\n",
    "\n",
    "### Two-parameter PDF\n",
    "\n",
    "### $f(x; \\alpha, \\beta) = \\frac{x^{\\alpha-1} e^{-x/\\beta}}{\\beta^\\alpha \\Gamma(\\alpha)}$\n",
    "\n",
    "### Cumulative Distribution Function (CDF)\n",
    "\n",
    "### $F(x; \\alpha, \\beta) = \\frac{1}{\\Gamma(\\alpha)} \\int_0^x t^{\\alpha-1} e^{-t/\\beta} \\, dt$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2287cab3",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "To better understand the Gamma Distribution, Suppose we have a manufacturing process that produces electronic components, and the time it takes for a component to fail follows a gamma distribution. This distribution is commonly used to model the lifespan or time to failure of various types of products.\n",
    "\n",
    "In this particular example, we will examine the gamma distribution with a shape parameter (α) of 2.5 implying moderate skewness, and a scale parameter (β) of 100 representing average failure time. By utilizing the gamma distribution, it is possible to determine probabilities linked to specific events in this scenario. For instance, we can calculate the likelihood of an element failing within its first 50 hours:\n",
    "\n",
    "### $\\alpha = 2.5$\n",
    "### $\\beta = 100$\n",
    "### $P(X \\leq 50) = \\int_0^{50} \\frac{1}{\\beta^\\alpha \\Gamma(\\alpha)} x^{\\alpha-1} e^{-x/\\beta} \\, dx = 0.03743422675270361$\n",
    "### $P(X > 200) = 1 - \\int_0^{200} \\frac{1}{\\beta^\\alpha \\Gamma(\\alpha)} x^{\\alpha-1} e^{-x/\\beta} \\, dx = 0.5494159513527802$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "b60d73ac",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 2.0), (0.01, 1.9603973466135105), (0.02, 1.9215788783046464), (0.03, 1.8835290671684974), (0.04, 1.8462326927732715), (0.05, 1.809674836071919), (0.06, 1.773840873434315), (0.07, 1.7387164707976117), (0.08, 1.7042875779324227), (0.09, 1.670540422822544)]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Gamma::new(1.0, 2.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..2000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Gamma::new(2.0, 2.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..2000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Gamma::new(9.0, 1.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..2000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points4:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = Gamma::new(7.5, 1.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..2000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "af3d674d",
   "metadata": {},
   "outputs": [
    {
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       "<text x=\"320\" y=\"5\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"Arial\" font-size=\"24.193548387096776\" opacity=\"1\" fill=\"#000000\">\n",
       "Gamma Distribution\n",
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       "<text x=\"527\" y=\"240\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝛼=9.0, 𝛽=1.0\n",
       "</text>\n",
       "<text x=\"527\" y=\"255\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝛼=7.5, 𝛽=1.0\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#FF0000\" stroke-width=\"1\" points=\"497,215 517,215 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#0000FF\" stroke-width=\"1\" points=\"497,230 517,230 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#00FF00\" stroke-width=\"1\" points=\"497,245 517,245 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"497,260 517,260 \"/>\n",
       "</svg>\n",
       "</div>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Gamma Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..20f64, 0f64..0.5f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"𝛼=1.0, 𝛽=2.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"𝛼=2.0, 𝛽=2.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"𝛼=9.0, 𝛽=1.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points4.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLACK\n",
    "    )).unwrap()\n",
    "    .label(\"𝛼=7.5, 𝛽=1.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLACK));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4df8326",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "320bf72c",
   "metadata": {},
   "source": [
    "## 9. Chi-squared Distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b747982f",
   "metadata": {},
   "source": [
    "### Probability Density Function (PDF)\n",
    "\n",
    "### $f(x; k) = \\frac{1}{{2^{\\frac{k}{2}} \\Gamma\\left(\\frac{k}{2}\\right)}} x^{\\frac{k}{2}-1} e^{-\\frac{x}{2}}$\n",
    "\n",
    "### Cumulative Distribution Function (CDF)\n",
    "\n",
    "### $F(x; k) = P(k/2, x/2) = \\gamma\\left(\\frac{k}{2}, \\frac{x}{2}\\right)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3cb89645",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "| | Car | Bicycle | Public Transport |\n",
    "| :--- | :--- | :--- | :--- |\n",
    "| Young | 45 | 20 | 35 |\n",
    "| Middle-aged | 55 | 40 | 25 \n",
    "| Senior | 30 | 15 | 25 |\n",
    "\n",
    "### Equation 1: Calculate the Chi-Squared Statistic.\n",
    "\n",
    "### $\\chi^2 = \\sum \\frac{(O_{ij} - E_{ij})^2}{E_{ij}}$\n",
    "### $\\chi^2 = \\frac{{(40 - 37.14)^2}}{{37.14}} + \\frac{{(30 - 35.71)^2}}{{35.71}} + \\frac{{(30 - 27.14)^2}}{{27.14}} + \\frac{{(20 - 22.86)^2}}{{22.86}} + \\frac{{(25 - 28.57)^2}}{{28.57}} + \\frac{{(25 - 22.86)^2}}{{22.86}}$\n",
    "### $\\chi^2 = 9.565073978309272$\n",
    "\n",
    "### Equation 2: Calculate the Degrees of Freedom\n",
    "\n",
    "### $\\text{Degrees of Freedom} = (r - 1) \\times (c - 1) = (3 - 1) \\times (3 - 1) = 4$\n",
    "\n",
    "### Equation 3: Calculate the p-value\n",
    "\n",
    "### $\\text{p-value} = 1 - \\text{CDF}(\\chi^2, \\text{Degrees of Freedom})$\n",
    "### $\\text{p-value} = 1 - \\text{{CDF}}(9.56, 4) = 0.04842715943407108$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "dc6c200d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, inf), (0.01, 3.969525474770118), (0.02, 2.792879016972343), (0.03, 2.2690027447147854), (0.04, 1.9552134698772794), (0.05, 1.7400739347725862), (0.06, 1.5805404178559017), (0.07, 1.455997867670957), (0.08, 1.3551684838810791), (0.09, 1.271292718201747)]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = ChiSquared::new(1.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = ChiSquared::new(4.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = ChiSquared::new(6.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points4:Vec<(f64,f64)> = {\n",
    "    let individuals_dist = ChiSquared::new(9.0).unwrap();\n",
    "    let mut individuals: Vec<f64> = (0..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = individuals.iter().map(|x| individuals_dist.pdf(*x)).collect();\n",
    "    individuals.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f2488c1c",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<div style=\"width:100%\"><svg width=\"640\" height=\"480\" viewBox=\"0 0 640 480\" xmlns=\"http://www.w3.org/2000/svg\">\n",
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       "<text x=\"320\" y=\"5\" dy=\"0.76em\" text-anchor=\"middle\" font-family=\"Arial\" font-size=\"24.193548387096776\" opacity=\"1\" fill=\"#000000\">\n",
       "Chi-squared Distribution\n",
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      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Chi-squared Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..10f64, 0f64..0.5f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"k=1.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"k=4.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"k=6.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points4.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLACK\n",
    "    )).unwrap()\n",
    "    .label(\"k=9.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLACK));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f36c1f5",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3f1970d",
   "metadata": {},
   "source": [
    "## 10. Beta Distribution\n",
    "\n",
    "### Probability Density Function (PDF)\n",
    "\n",
    "### $f(x; \\alpha, \\beta, a, b) = \\frac{{(x - a)^{\\alpha - 1}(b - x)^{\\beta - 1}}}{{B(\\alpha, \\beta)(b - a)^{\\alpha + \\beta - 1}}}$\n",
    "\n",
    "### Standard Beta Distribution PDF (a = 0, b = 1)\n",
    "\n",
    "### $f(x; \\alpha, \\beta) = \\frac{{x^{\\alpha - 1}(1 - x)^{\\beta - 1}}}{{B(\\alpha, \\beta)}}$\n",
    "\n",
    "### Cumulative Distribution Function (CDF)\n",
    "\n",
    "### $F(x; \\alpha, \\beta, a, b) = \\frac{{B_x(\\alpha, \\beta)}}{{B(\\alpha, \\beta)}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bac850b1",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "To better understand the Beta distribution, Suppose we are analyzing the effectiveness of a new drug in treating a certain medical condition. The response rate to the drug follows a beta distribution with shape parameters α = 4 and β = 3. We are interested in determining the probability that the drug achieves a response rate between 60% and 70%. To compute the probability of the drug achieving a response rate between 60% and 70%, we can use the following equations:\n",
    "\n",
    "- Step 1: Compute the cumulative distribution function (CDF) values at the upper and lower bounds:\n",
    "\n",
    "### $F(x;\\alpha,\\beta) = \\int_{0}^{x} t^{\\alpha-1}(1-t)^{\\beta-1} dt$\n",
    "\n",
    "- Step 2: Substitute the values for the lower and upper bounds, as well as the shape parameters:\n",
    "\n",
    "### $F(0.7;4,3) - F(0.6;4,3)$\n",
    "\n",
    "- Step 3: Calculate the integral using the beta CDF formula:\n",
    "\n",
    "### $F(0.7;4,3) = \\int_{0}^{0.7} t^{4-1}(1-t)^{3-1} dt = 0.74431$\n",
    "### $F(0.6;4,3) = \\int_{0}^{0.6} t^{4-1}(1-t)^{3-1} dt = 0.54432$\n",
    "\n",
    "- Step 4: Calculate the probability by subtracting the lower CDF from the upper CDF:\n",
    "\n",
    "### $P(0.6 \\leq x \\leq 0.7) = F(0.7;4,3) - F(0.6;4,3)$\n",
    "### $P(0.6 \\leq x \\leq 0.7) = 0.74431 - 0.54432$\n",
    "\n",
    "So, the probability of the drug achieving a response rate between 60% and 70% is 0.19999."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "80a4622d",
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let dist = Beta::new(0.5, 0.5).unwrap();\n",
    "    let mut trials: Vec<f64> = (0..100).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let dist = Beta::new(5.0, 1.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (0..100).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let dist = Beta::new(2.0, 2.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (0..100).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points4:Vec<(f64,f64)> = {\n",
    "    let dist = Beta::new(9.0, 5.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (0..100).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "f73ba601",
   "metadata": {},
   "outputs": [
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       "Beta Distribution\n",
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       "0.6\n",
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       "0.7\n",
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       "0.8\n",
       "</text>\n",
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       "0.9\n",
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       "1.0\n",
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       "α = β = 0.5\n",
       "</text>\n",
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       "α = β = 0.5\n",
       "</text>\n",
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       "α = 5.0, β = 1.0\n",
       "</text>\n",
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       "α = β = 0.5\n",
       "</text>\n",
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       "α = 5.0, β = 1.0\n",
       "</text>\n",
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       "α = 2.0, β = 2.0\n",
       "</text>\n",
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       "α = β = 0.5\n",
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       "α = 5.0, β = 1.0\n",
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       "α = 2.0, β = 2.0\n",
       "</text>\n",
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       "α = 9.0, β = 5.0\n",
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      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Beta Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..1f64, 0f64..5.02f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"α = β = 0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"α = 5.0, β = 1.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"α = 2.0, β = 2.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points4.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLACK\n",
    "    )).unwrap()\n",
    "    .label(\"α = 9.0, β = 5.0\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLACK));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c53d5831",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96c3637d",
   "metadata": {},
   "source": [
    "## 11. Negative Binomial Distribution\n",
    "\n",
    "### $P(x,r,p) = \\binom{x-1}{r-1} \\cdot p^r \\cdot (1-p)^{x-r}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ac8fbb2",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "To better understand this distribution, suppose we are conducting a series of independent coin flips and want to determine the probability of obtaining the third consecutive heads on the seventh flip. Let's calculate this probability using the negative binomial distribution.\n",
    "\n",
    "Consider this example: the likelihood of obtaining heads on each toss (p) is 0.5, and our focus lies in achieving three successes overall (r) during seven consecutive flips (x).\n",
    "\n",
    "- Step 1: Define the variables:\n",
    "\n",
    "Probability of success (p): 0.5\n",
    "Number of trials (x): 7\n",
    "Number of successes (r): 3\n",
    "\n",
    "- Step 2: Calculate the negative binomial probability.\n",
    "\n",
    "Substitute the values into the formula and calculate the probability.\n",
    "\n",
    "### $P(x=7,r=3,p=0.5) = \\binom{7-1}{3-1} \\cdot 0.5^3 \\cdot (1-0.5)^{7-3}$\n",
    "### $P(x=7,r=3,p=0.5) = \\binom{6}{2} \\cdot 0.5^3 \\cdot 0.5^4$\n",
    "### $P(x=7,r=3,p=0.5) = \\frac{6!}{2! \\cdot (6-2)!} \\cdot 0.5^3 \\cdot 0.5^4$\n",
    "### $P(x=7,r=3,p=0.5) = \\frac{6!}{2! \\cdot 4!} \\cdot 0.5^7$\n",
    "### $P(x=7,r=3,p=0.5) = \\frac{6 \\cdot 5}{2} \\cdot 0.5^7 $\n",
    "### $P(x=7,r=3,p=0.5) = 0.03515625000000001$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "339e63fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(0.0, 0.5000000000000002), (1.0, 0.2500000000000001), (2.0, 0.12500000000000008), (3.0, 0.06250000000000003), (4.0, 0.031250000000000014), (5.0, 0.01562500000000002), (6.0, 0.007812500000000002), (7.0, 0.003906250000000001), (8.0, 0.001953125), (9.0, 0.0009765625)]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let dist = NegativeBinomial::new(1.0, 0.5).unwrap();\n",
    "    let mut trials: Vec<f64> = (0 .. 25).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pmf(*x as u64)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let dist = NegativeBinomial::new(2.0, 0.5).unwrap();\n",
    "    let mut trials: Vec<f64> = (0 .. 25).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pmf(*x as u64)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let dist = NegativeBinomial::new(4.0, 0.5).unwrap();\n",
    "    let mut trials: Vec<f64> = (0 .. 25).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pmf(*x as u64)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points4:Vec<(f64,f64)> = {\n",
    "    let dist = NegativeBinomial::new(5.0, 0.5).unwrap();\n",
    "    let mut trials: Vec<f64> = (0 .. 25).map(f64::from).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pmf(*x as u64)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "dd4b14e0",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Negative Binomial Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(0f64..15f64, 0f64..0.6f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"r=1.0, p=0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"r=2.0, p=0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"r=5.0, p=0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points4.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLACK\n",
    "    )).unwrap()\n",
    "    .label(\"r=1.0, p=0.5\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLACK));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9eb800c6",
   "metadata": {},
   "source": [
    "<hr />"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa75545c",
   "metadata": {},
   "source": [
    "## 12. Laplace Distribution\n",
    "\n",
    "### Probability Density Function (PDF)\n",
    "\n",
    "### $f(x|\\lambda) = \\frac{1}{2\\lambda} e\\left(-\\frac{|x|}{\\lambda}\\right)$\n",
    "\n",
    "### Cumulative Distribution Function (CDF)\n",
    "\n",
    "### $F(x|\\lambda) = \\begin{cases}\n",
    "\\frac{1}{2} e\\left(\\frac{x}{\\lambda}\\right) & \\text{if } x < 0 \\\\\n",
    "1 - \\frac{1}{2} e\\left(-\\frac{x}{\\lambda}\\right) & \\text{if } x \\geq 0\n",
    "\\end{cases}$\n",
    "\n",
    "### PDF for the two-parameter Laplace distribution\n",
    "\n",
    "### $f(x|\\lambda,\\mu) = \\frac{1}{2\\lambda} e\\left(-\\frac{|x-\\mu|}{\\lambda}\\right)$\n",
    "\n",
    "### CDF for the two-parameter Laplace distribution\n",
    "\n",
    "### $F(x|\\lambda,\\mu) = \\begin{cases}\n",
    "\\frac{1}{2} e\\left(\\frac{x-\\mu}{\\lambda}\\right) & \\text{if } x < \\mu \\\\\n",
    "1 - \\frac{1}{2} e\\left(-\\frac{x-\\mu}{\\lambda}\\right) & \\text{if } x \\geq \\mu\n",
    "\\end{cases}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "150723b8",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "Suppose we have a manufacturing process that produces electronic components. The time it takes for a component to fail follows a Laplace distribution with a location parameter μ=10 and a scale parameter λ=2. We want to compute the probability that a component will fail in the [8, 12] hours interval after production. The Laplace distribution probability density function (PDF) is given by:\n",
    "\n",
    "### $f(x|\\mu,\\lambda) = \\frac{1}{2\\lambda} e\\left(-\\frac{|x-\\mu|}{\\lambda}\\right)$\n",
    "\n",
    "To find the cumulative probability, we integrate the PDF from negative infinity to a given value x:\n",
    "\n",
    "### $F(x|\\mu,\\lambda) = \\int_{-\\infty}^{x} f(t|\\mu,\\lambda) dt$\n",
    "\n",
    "The cumulative distribution function is defined as follows:\n",
    "\n",
    "### $F(x|\\mu,\\lambda) = \\begin{cases}\n",
    "\\frac{1}{2} e\\left(\\frac{x-\\mu}{\\lambda}\\right) & \\text{if } x < \\mu \\\\\n",
    "1 - \\frac{1}{2} e\\left(-\\frac{x-\\mu}{\\lambda}\\right) & \\text{if } x \\geq \\mu\n",
    "\\end{cases}$\n",
    "\n",
    "Given our parameters of μ=10 and λ=2, we aim to determine the probability that the component will malfunction within a time frame spanning from 8 to 12 hours. This can be mathematically represented as:\n",
    "\n",
    "### $P(8 \\leq X \\leq 12) = F(12|\\mu,\\lambda) - F(8|\\mu,\\lambda)$\n",
    "\n",
    "Substituting the values into the equation, we have:\n",
    "\n",
    "### $P(8 \\leq X \\leq 12) = \\left(1 - \\frac{1}{2} e\\left(-\\frac{12-10}{2}\\right)\\right) - \\left(\\frac{1}{2} e\\left(\\frac{8-10}{2}\\right)\\right)$\n",
    "\n",
    "Simplifying further:\n",
    "\n",
    "### $P(8 \\leq X \\leq 12) = \\left(1 - \\frac{1}{2} e(-1)\\right) - \\left(\\frac{1}{2} e(-1)\\right)$\n",
    "\n",
    "Finally, we evaluate the expression to obtain the result:\n",
    "\n",
    "### $P(8 \\leq X \\leq 12) \\approx 0.6321205588285577$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "0502467a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-10.0, 2.2699964881242427e-5), (-9.99, 2.2928103321103654e-5), (-9.98, 2.315853459040381e-5), (-9.97, 2.339128173246185e-5), (-9.96, 2.3626368022185935e-5), (-9.95, 2.3863816968400985e-5), (-9.94, 2.4103652316199415e-5), (-9.93, 2.4345898049315906e-5), (-9.92, 2.4590578392525647e-5), (-9.91, 2.483771781406686e-5)]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "let random_points1:Vec<(f64,f64)> = {\n",
    "    let dist = Laplace::new(0.0, 1.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (-1000..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points2:Vec<(f64,f64)> = {\n",
    "    let dist = Laplace::new(0.0, 2.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (-1000..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points3:Vec<(f64,f64)> = {\n",
    "    let dist = Laplace::new(0.0, 4.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (-1000..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "let random_points4:Vec<(f64,f64)> = {\n",
    "    let dist = Laplace::new(-5.0, 4.0).unwrap();\n",
    "    let mut trials: Vec<f64> = (-1000..1000).map(|x| x as f64 / 100.0).collect();\n",
    "    let y_iter: Vec<f64> = trials.iter().map(|x| dist.pdf(*x)).collect();\n",
    "    trials.into_iter().zip(y_iter).collect()\n",
    "};\n",
    "random_points1[..10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "e810c8d3",
   "metadata": {},
   "outputs": [
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       "<rect x=\"503\" y=\"200\" width=\"132\" height=\"75\" opacity=\"1\" fill=\"#FFFFFF\" stroke=\"none\"/>\n",
       "<rect x=\"503\" y=\"200\" width=\"132\" height=\"75\" opacity=\"1\" fill=\"none\" stroke=\"#000000\"/>\n",
       "<text x=\"543\" y=\"210\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝜇=0 ,𝜆=1\n",
       "</text>\n",
       "<text x=\"543\" y=\"225\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝜇=0 ,𝜆=2\n",
       "</text>\n",
       "<text x=\"543\" y=\"240\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝜇=0 ,𝜆=4\n",
       "</text>\n",
       "<text x=\"543\" y=\"255\" dy=\"0.76em\" text-anchor=\"start\" font-family=\"sans-serif\" font-size=\"9.67741935483871\" opacity=\"1\" fill=\"#000000\">\n",
       "𝜇=-5 ,𝜆=4\n",
       "</text>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#FF0000\" stroke-width=\"1\" points=\"513,215 533,215 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#0000FF\" stroke-width=\"1\" points=\"513,230 533,230 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#00FF00\" stroke-width=\"1\" points=\"513,245 533,245 \"/>\n",
       "<polyline fill=\"none\" opacity=\"1\" stroke=\"#000000\" stroke-width=\"1\" points=\"513,260 533,260 \"/>\n",
       "</svg>\n",
       "</div>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evcxr_figure((640, 480), |root| {\n",
    "    _ = root.fill(&WHITE);\n",
    "    let mut chart = ChartBuilder::on(&root)\n",
    "        .caption(\"Laplace Distribution\", (\"Arial\", 30).into_font())\n",
    "        .x_label_area_size(40)\n",
    "        .y_label_area_size(40)\n",
    "        .build_cartesian_2d(-10f64..10f64, 0f64..0.6f64)?;\n",
    "    \n",
    "    chart.configure_mesh()\n",
    "        .x_desc(\"Trials\")\n",
    "        .y_desc(\"Probability\")\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points1.iter().map(|(x, y)| (*x, *y)),\n",
    "        &RED\n",
    "    )).unwrap()\n",
    "    .label(\"𝜇=0 ,𝜆=1\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &RED));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points2.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLUE\n",
    "    )).unwrap()\n",
    "    .label(\"𝜇=0 ,𝜆=2\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLUE));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points3.iter().map(|(x, y)| (*x, *y)),\n",
    "        &GREEN\n",
    "    )).unwrap()\n",
    "    .label(\"𝜇=0 ,𝜆=4\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &GREEN));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    chart.draw_series(LineSeries::new(\n",
    "        random_points4.iter().map(|(x, y)| (*x, *y)),\n",
    "        &BLACK\n",
    "    )).unwrap()\n",
    "    .label(\"𝜇=-5 ,𝜆=4\")\n",
    "    .legend(|(x,y)| PathElement::new(vec![(x,y), (x + 20,y)], &BLACK));\n",
    "\n",
    "    chart.configure_series_labels()\n",
    "        .background_style(&WHITE)\n",
    "        .border_style(&BLACK)\n",
    "        .draw()?;\n",
    "\n",
    "    Ok(())\n",
    "}).style(\"width:100%\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bbbb1f75",
   "metadata": {},
   "source": [
    "---\n",
    "---"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Rust",
   "language": "rust",
   "name": "rust"
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   "file_extension": ".rs",
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