Nahodna_velicina_spojita.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Materiály vznikají průběžně a jsou bez záruky - prosím o report chyb :-)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats.distributions import uniform, expon, norm\n",
    "import matplotlib.pylab as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Téma 4: Spojité náhodné veličiny"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Už víme, že náhodné veličiny mohou z hlediska oboru hodnot být\n",
    "\n",
    "- **diskrétní** - nabývají hodnot z nejvýše spočetné množiny $\\{x_1,x_2,\\ldots\\}$. Teoreticky můžeme pro všechna $k=1,2,\\ldots$ kvantifikovat pravděpodobnost jednotlivých hodnot $P(X=x_k)$.\n",
    "- **spojité** - veličina $X$ nabývá nespočetně mnoha možných hodnot, žije na nějakém kontinuu, např. množině reálných čísel $\\mathbb{R}$ či libovolné její podmnožině (intervalu). Nelze pro hodnoty $x$ kvantifikovat jejich pravděpodobnost $P(X=x)$, neboť (i) takových bodů je v kontinuu nekonečno, ale pravděpodobnosti se musejí nakumulovat do jedničky a (ii) bod kontinua má míru nula (vzpomeňte na definici náhodné veličiny). Jde to podchytit i intuitivně - jaká je pravděpodobnost, že vybraná slečna bude mít 1.7 metru s nekonečnou přesností?\n",
    "\n",
    "*Kdybychom uměli trochu více s mírou, řekli bychom si, že u diskrétních veličin používáme aritmetickou a u spojitých Lebesgueovu a operace by nám zůstaly pro oba případy stejné, protože to je právě míra, která říká, jak s tím naložit...*\n",
    "\n",
    "**Úkol: vymyslete pár příkladů diskrétních a spojitých veličin.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> **Definice**\n",
    ">\n",
    "> Náhodná veličina $X$ s hodnotami $x$ se nazývá spojitá, jestliže existuje nezáporná funkce $f_X(x)$ taková, že pro každé $x\\in\\mathbb{R}$ můžeme distribuční funkci $F_X(x)$ vyjádřit jako\n",
    ">\n",
    "> $$\n",
    "F_X(x) = P(X\\leq x) = \\int_{-\\infty}^x f_X(t)dt.\n",
    "$$\n",
    "\n",
    "Funkce $f_X$ se nazývá hustotou pravděpodobnosti náhodné veličiny $X$. Platí, celkem zřejmě, že\n",
    "\n",
    "$$\n",
    "f_X(x) = \\frac{dF_X(x)}{dx}.\n",
    "$$\n",
    "\n",
    "Podíváme-li se na hustotu a distribuční funkci, jakou hodnotu bude mít následující integrál?\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^{\\infty} f_X(x) dx\n",
    "$$\n",
    "\n",
    "(samozřejmě 1, je to normalizační podmínka: pravděpodobnosti se sčítají/integrují do jedné)\n",
    "\n",
    "*Vzpomeňte - integrál je vlastně něco, jako \"suma přes kontinuum\", nasčítává maličké sloupečky v grafu. Zde je vidět ta aplikace míry - v diskrétním případě sčítáme, ve spojitém také sčítáme, ale kvůli kontinuu si s obyčejnou sumou nevystačíme. A nebo naopak, ta suma ve spojitém případě je také integrál, jen s jinou mírou :)*\n",
    "\n",
    "![cdf](img/cdf.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stejně, jako v diskrétním případě, i zde tedy platí, že distribuční funkce jsou \"nakumulované\" pravděpodobnosti. A celkem logicky platí i pravidla pro počítání pravděpodobnosti nějaké množiny, $P(X\\in(a,b])$. Zamyslete se, jak by to bylo s intervaly $[a,b], (a,b), [a,b)$.\n",
    "\n",
    "![cdf2](img/cdf-pst.jpg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hustota pravděpodobnosti a distribuční funkce spojité veličiny\n",
    "\n",
    "Hustota pravděpodobnosti říká, jak je pravděpodobnost rozdělena mezi hodnoty náhodné veličiny. Neříká ale, jaká je pravděpodobnost těchto veličin - tento rozdíl je nutné mít na paměti.\n",
    "\n",
    "Pamatujete si, jak jsme z empirických četností a kumulativních empirických četností přecházeli v limitě k pravděpodobnostní a distribuční funkci? Tady je to stejné, jen bychom malovali histogram a kumulativní histogram a zmenšovali šířku obdélníčků. V limitě bychom se dostali k příslušným funkcím :)\n",
    "\n",
    "Podívejme se na to názorně - který obrázek vyjadřuje kterou funkci?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1080x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "nbins = 30\n",
    "plt.figure(figsize=(15,3))\n",
    "x = norm.rvs(size=100000)\n",
    "plt.subplot(1,2,1)\n",
    "plt.hist(x, bins=nbins, density=True, color='orange')\n",
    "xx = np.linspace(-5,5)\n",
    "plt.plot(xx, norm.pdf(xx), color='blue', lw=3)\n",
    "plt.subplot(1,2,2)\n",
    "plt.hist(x, bins=nbins, density=True, cumulative=True, color='orange')\n",
    "xx = np.linspace(-5,5)\n",
    "plt.plot(xx, norm.cdf(xx), color='blue', lw=3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "**Spočtěte příklad 5.1 dole.**\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Funkce přežití\n",
    "\n",
    "Zatímco distribuční funkce říká, jaká je pravděpodobnost, že veličina $X$ bude mít hodnotu $x$ a menší, tj.\n",
    "\n",
    "$$\n",
    "F_X(x) = P(X\\leq x),\n",
    "$$\n",
    "\n",
    "funkce přežití říká pravý opak,\n",
    "\n",
    "$$\n",
    "S_X(x) = 1 - F_X(x) = P(X > x).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Momenty\n",
    "\n",
    "Už víme, že integrál je \"suma přes kontinuum\", tak porovnejme diskrétní a spojité momenty, ať vidíme, že to je snadné a přímočaré. Pro jednoduchost budeme v diskrétním světě psát $P(X=x_k) = p_k$, tedy $p_k$ je pravděpodobnost hodnoty $x_k$.\n",
    "\n",
    "**Střední hodnota:**\n",
    "\n",
    "$$\n",
    "EX = \\sum_k x_k \\cdot p_k \\qquad \\longleftrightarrow \\qquad EX = \\int_{\\mathbb{R}} x \\cdot f(x) dx,\n",
    "$$\n",
    "\n",
    "**Rozptyl:**\n",
    "\n",
    "$$\n",
    "\\operatorname{var} X = \\sum_k (x_k-EX)^2 \\cdot p_k \\qquad \\longleftrightarrow \\qquad \n",
    "\\operatorname{var} X = \\int_{\\mathbb{R}} (x - EX)^2 \\cdot f(x) dx.\n",
    "$$\n",
    "\n",
    "V obou případech platí, že $\\operatorname{var}X = EX^2 - (EX)^2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tabulka pro přehled\n",
    "\n",
    "| | Diskrétní náhodná veličina | Spojitá náhodná veličina |\n",
    "|---|---|---|\n",
    "| | pravděpodobnostní funkce $p_k=P(X=x_k)$| hustota pravděpodobnosti $f(x)$ |\n",
    "|$EX$| $\\sum_{k} x_k p_k$ | $\\int_{\\mathbb{R}} x f(x) dx$ |\n",
    "|$\\operatorname{var}X$ | $\\sum_k (x_k - EX)^2 p_k$ | $\\int_{\\mathbb{R}} (x-EX)^2 f(x) dx$ |\n",
    "|normalizační podmínka | $\\sum_k p_k = 1$ | $\\int_{\\mathbb{R}} f(x)dx =1$|\n",
    "\n",
    "*A kdybychom uměli míru, vůbec bychom se nějakými rozdíly nemuseli zabývat :)*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Hustoty a distribuční funkce - pár obrázků\n",
    "\n",
    "Rovněž se koukněme na některé hustoty, distribuční funkce a střední hodnoty graficky:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x360 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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TauojXDd3MZFolMeuOZmxA7XVWES6hp3ltdxTuJa5b75PfSTKzCMHccWHRnHymH7ayBwANW0iHaisup6bnnyLF9/ezsePGcQv/+s4euVmBx1LRJpx5z/WsHLrHu6/bKoaNhHpEmrqI9xTuI7f//s96iJRPjtlGNefPpYR/boHHa1LU9Mm0kFWbC5j1tzFbCmt5ntnH8UXTzlcW6pE0tim4irue2U9500eysyjBgUdR0Sk3f3z3e38cN7bbCqu5uxjB/P1jx6h89LShJo2kXbm7jw6fxM/evZt+nbvxuNfOpHjR/YNOpaItOCOv68hIwO+eeb4oKOIiLSrsup6fvjMCv68dAtjBvTgkaumc/LY/kHHkgRq2kTaUVVdA9/70wqeXrKZU8f1547PTaJffk7QsUSkBTvKa5j31mYumT6Swb3zgo4jItJuXl+3m68/sZTt5bXcMHMc158+VpfoT0Nq2kTaydod5cyau5g1Oyr46keOYPYZY8nM0OGQImHw8Bvv0xB1Ljt5VNBRRETahbvzm5fX8csXVzGqXw+evPYkJo/oE3QsaYaaNpF28MzSzdz89HLysjN56MrpnDJOhxiIhIW78/TiIk4Z25/D+/cIOo6ISMpV1jZw05PL+MvyrXzquCH84vyJuk9smtP/jkgK1TZE+Mlz7/DwG+8zdWQf7rp4Cof1zg06loi0wtJNpRSVVHPDzHFBRxERSbmtZdVc8ccFrN5eznfOOpKrTx2tC6OFgJo2kRTZVFzFrLmLWb65jGtOG803Pz6e7EwdEy4SNs8t20q3zAw+dsxhQUcREUmpNdvLuewP8ymvaWDOldM4ddyAoCNJktS0iaTA39/ZzteeWIoD9156vFb2RELsn+/u4OSx/eidp3soikjnsWhjMVc+sJBuWRk8/qWTOHpIr6AjSSuoaRNpg4ZIlF/+bRW/e/k9JgztxT0XH6+bT4qEWG1DlPW7Kvn8iSODjiIikjKvrtnFVQ8uYHDvPB68chrD+2pdJWzUtIkcou17avjyI0uYv6GYi6eP4AefPJrc7MygY4lIG5RV1wNwiu5PJCKdxH/W7uKLcxZweP8ePHzVdPrr1kOhpKZN5BD8Z+0ubnhsCZW1Ee743CQ+PXlo0JFEJAXKquvpn5/DEYPyg44iItJmryU0bI9cfSJ9e3QLOpIcIjVtIq0QjTp3/2stt/99NaMH5PPo1VMYN6hn0LFEJEXKa+o5cXRfXUlNRELv9XW7uXLOAkb168Hcq6arYQs5NW0iSSqurOOrjy/l5dU7+fSkIfzsMxPpkaOvkEhnUReJUtcQZdLwgqCjiIi0yYrNZVz94EKG9+nO3Kum00+HRIae1jhFkrBoYwmzH1nM7oo6fvrpCVwyfYS2xIt0MpW1DQAcO0xNm4iE18bdlVz+xwX0zsvmoS+qYess1LSJHIS784f/bODnz69kcEEuT113MhOH9Q46loi0g4raCADH6DLYIhJSO8trufT++USiUeZceSKH9c4NOpKkiJo2kWbsqannW08u44UV2/jo0YP41WePo3d33bdJpLOqrG0gr1umDnsWkVAqr6nn8j/OZ2d5LY9cPZ2xA3VBpc5Ef5lEmvDOlj3MmruITSXVfOesI7n61NE6HFKkk6usa9ANtUUklBoiUWbNXcyqbeXcd9lUJo/oE3QkSTE1bSIJ3J0nFm7iB8+8TUH3bB675kROGNU36Fgi0s721NRT3xAlT/daFJEQ+slz7/DKml384vyJzBg/MOg40g7UtInEVddF+N6fV/DU4iI+NLYfv75wsm5AKdJFrNtRAUBeNzVtIhIuD72+gTmvb+TqUw/ncyeMCDqOtBM1bSLAup0VzHp4Mat3lPOVmeO4YeY4MjN0OKRIV7FmRwUjQHvaRCRUXlmzkx89+w4zjxzItz9xVNBxpB2paZMu79m3tvDtp5aRk53JnCumcdoRA4KOJCIdbN2OCkaakZulpk1EwmHtjgpmzV3M2AH5/PqiydrY3MmpaZMuq7Yhwi1/Wcmc1zcyZUQBd18yhcG984KOJSIBWLujgrOzM9H1hkQkDEoq67hqzgK6ZWZw32VTyddVbzu9jGQmMrMzzWyVma01s283Mf5yM9tpZkvjP1elPqpI6hSVVHHBb19nzusb+eIph/P4l05Sw9bJqG5Ja6zdWUGuzmeTgKluSTIcuG7uIraU1vC7S49neN/uQUeSDtBiW25mmcDdwEeBImCBmc1z93caTfq4u89uh4wiKVVR75x956tEo85vPz+FMycMDjqSpJjqlrRGQyTK5pJqcrOS2o4p0i5UtyQZDmyrjPLGe8XcdsFxTNUVrruMZP5CTQPWuvt77l4HPAac276xRFKvIRLl/eIqisqjDC3I49kvn6KGrfNS3ZKkbS+vpSHq5OgiJBIs1S1p0bayGspqnVkzxnDelGFBx5EOlMwBsEOBTQnPi4DpTUx3vpmdBqwGvurumxpPYGbXANcADBo0iMLCwlYHTpWKiopAl58s5UyN0poov3mrlv8uraZXN7hxQgMbVixgQ9DBDiLd39M0p7oVoLDkhFjW5/75GgANNVWURqpZmobZw/KehiVnmkpZ3YL0qV1h+kyke9ZlOxuYtruSHtkwNWcrhYXbgo50UOn+fu4VlpzJNG1NnZbtjZ4/Czzq7rVmdi0wBzjjgBe53wvcCzB16lSfMWNG69KmUGFhIUEuP1nK2Xavr9vNTx9dQkUtjBmQT3akmqNnnh50rBal83saAqpbAQpLTohl7d9zLMx/i74FvcjLzkzL7GF5T8OSM02lrG5B+tSuMH0m0jnr6u3lzP7Xa8zolsXQHs7E07UekyphyZnM4ZFFwPCE58OALYkTuPtud6+NP/09cHxq4okcumjUuftfa7nkvjfolZfFM9efwoCeull2F6G6JUkrKqkGIEfntEmwVLekScWVdXxxzgJyszMZf1hPdGX/rimZv1ALgHFmdriZdQMuBOYlTmBmiScGnQOsTF1EkdYriRe4X764irOPHcK82acw/rCeQceSjqO6JUnbVFLFoF45ZOh6/xIs1S05QF1DlGsfWsT2PbX8/gvHa+NSF9bi4ZHu3mBms4EXgUzgD+7+tpn9GFjo7vOAr5jZOUADUAxc3o6ZRQ5qyfslzH5kCTvLa/nJucfw+RNHYloZ61JUt6Q1ikqqGNZHl8yWYKluSWPuznf/tJz5G4r59YWTmDyiT9CRJEBJ3YnP3Z8Hnm807AcJj28Gbk5tNJHWcXfmvLaBnz2/koE9c3nyupM4dlhB0LEkIKpbkqyikmqOH6mVIQme6pYkuu+V9fy/RUV85YyxnDtpaNBxJGC6fbp0CuU19Xz7qeX8ZflWZh45kP+94DgKuncLOpaIpLmoO9vKahhSkBd0FBGRff6xcju3vLCSsyYexo0fOSLoOJIG1LRJ6K3cuodZcxfzfnEV3zrzSL502mgydJauiCShog4aos4gXaRIRNLEO1v28JVHlzBhSG/+978maZ1GADVtEnJPLNzE9/+8gt552Txy1XSmj+4XdCQRCZHS2igAA3vlBpxERAQ2l1ZzxQPz6ZWXze+/MJW8bplBR5I0oaZNQqmmPsIPnlnBEwuLOGl0P+68aLIu5y8irVZaG7sN1kDVDxEJWFlVPZf9YT5VdRGevPZkDuutjUnyATVtEjrrd1Vy3cOLeHdbObNPH8tXP3oEmTp0QEQOQdm+pk0rRyISnJr6CFc/uJD3d1cx58ppuk2RHEBNm4TK88u3ctOTy8jKNP54xQmcPn5g0JFEJMT27WnrpT1tIhKMSNT52hNLmb+hmDsvmsxJY3SqhxxITZuEQl1DlFueX8kDr21g0vAC7r5kCkN1tTcRaaPSWqdnbha52TpvREQ6XjTqfOupZTy/fBvfO/sozjluSNCRJE2paZO0t7m0muvnLmbpplKu+NAobv7EUXTLygg6loh0AmW1rkMjRSQQ7s4P573Nk4uKuPEj47jq1NFBR5I0pqZN0tq/Vu3gq48vpSHi3H3xFM4+dnDQkUSkEymtdQb2VdMmIh3L3bnl+ZU89MZGvvTh0dwwc1zQkSTNqWmTtBSJOre/tJq7/rWWIw/ryT2XTGH0gPygY4lIJ1NW6xyj89lEpAO5O7e+8C6/f2U9l500km+feSRmuqCaHJyaNkk7O8pruOHRpbz+3m4umDqMH587QeebiEjKuXv88Eg1bSLSMaJR5/vPrGDum+9z6Ykj+eGnjlHDJklR0yZp5Y33dvPlR5ewp7qe//nssVwwdXjQkUSkk6qqi1AXhf75atpEpP01RKJ888ll/GnJZq798Bi+deZ4NWySNDVtkhaiUee3/17Hr15cxch+PXjwymkcNbhX0LFEpBMrrqwDoE+PbgEnEZHObk9NPdfPXcwra3bxjY8dwfWnj1XDJq2ipk0CV1pVx9efeIt/vLuDsycO5tbzJ9IzNzvoWCLSyZVW1QPQp7uaNhFpP5uKq7jygQWs31XJredN5MJpI4KOJCGkpk0C9damUmbNXcyO8hp+9KmjuezkUdryJCIdorgqtqetbw9tJBKR9vHGe7uZ/chi6hqiPHjlNE4e2z/oSBJSatokEO7OQ29s5KfPrWRAzxye+NJJTB7RJ+hYItKFlMabtgLtaRORFItGnXsK13LbS6sZ1a8H935hKmMH6irYcujUtEmHq6ht4Oanl/PsW1s4ffwAbrtgks4pEZEOt/ectr5q2kQkhbaWVXPTk8t4Zc0uzjluCLecN5H8HK1yS9voEyQdatW2cq6bu4gNuyr55sfHc92Hx5CRocMhRaTjlVTVY0CvPB0eKSJt5+48On8TP39+JQ1R55bPTOSiacN12oekhJo26TBPLSriu39eTn5ONg9fNZ2Tx+i4bhEJTkllHT2yIVMbjkSkjVZsLuMnz73Dm+uLOWl0P35x/rGM6Nc96FjSiahpk3ZXUx/hR/Pe5rEFm5h+eF/+76LJDOyVG3QsEeniSqrqyM9WwyYih25LaTW3v7SaJxcXUZCXzS2fmciFJwzXUUSScmrapF1t2FXJrLmLeWfrHmbNGMPXPnoEWZkZQccSEYk1bd20YiUirffezgp+9/J7PL2kCMO45tTRzDp9LL11uLW0EzVt0m7+umIr3/x/y8jIMO6/bCozjxoUdCQRkX1KKuu1p01EklbXEOUfK7fz2IJN/HvNTrplZnDxtBFcfdpohvXRoZDSvtS0ScrVR6Lc+sK73P/qeo4b1pu7Lp7C8L4qZiKSXkqq6hiTr6ZNRJpXUx/h9XW7+ds72/jb29vZXVnH4N65fPmMcVx64kgG9MwJOqJ0EUk1bWZ2JvBrIBO4z91vbTQ+B3gQOB7YDXzO3TekNqqEwZbSamY/spjF75dy2Ukj+c7ZR5GTlRl0LOmCVLekJSVVdeT3UX2S9KG6Fbya+gjLispYsKGY+euLWbihmMq6CPk5WcwYP4Dzjx/GaeMG6AJG0uFabNrMLBO4G/goUAQsMLN57v5OwmRfBErcfayZXQj8AvhcewSW9PXy6p3c+NgS6hqi/N9Fk/nUcUOCjiRdlOqWtKS6LkJNfZT8bmraJD2obnWc+kiUbWU1bC2rYUtpNet3VbJ6ezlL1lex48W/EvXYdOMG5vOZKUP5yFGDOGlMP22ElkAls6dtGrDW3d8DMLPHgHOBxCJyLvCj+OMngbvMzNzd2xKuPhJl3c6K/YY1N8emhjsHDtw73cY9EVZsLmt22a2d3/7TNTW/Jl6bxHLXlkToubG4Dcs99CzNL+fAgU+truO5F+dzxMCe3PP5KYwZkN/MHEU6RGB1C2InqNdH2jybAxSVR1m1rTzl8021MOTcUlYNQEGOtpZL2gi0bm0tq6a0qh744G+/4/utByQO3/vc943zhMf7XsGakgj5G4rxxNfHH3jjee57nNyyIhGntiFKbUOEuobovse19bHHVXURyqrrKauui/9bT0lVPbsqavf7vcxgVL8eDM3P4ILphzNhaG+mjupL3x7dDum9FGkPyTRtQ4FNCc+LgOnNTePuDWZWBvQDdrUl3O6KOs6845W2zOLgXnu1/eadSm++HnSCpJw/ZRg//fQE8rTlWoIXWN0CuPKBBWzYXdXW2TTtP/9un/mmWkhyDu6hq9kswqRSAAAF80lEQVRK2gi0bt3x0hoeX7ip5QkPRQDrMRkG3btl0Tsve9/P6P75FHTPZmCvXIYW5DKkII/BvfMY1ieP3OxMCgsLmTFjfIdnFUlGMk1bU5shG2/RSWYazOwa4BqAQYMGUVhYeNAF10Wc6ycdeIJnc9tFk73hvAE1NTXk5n5wr7DWvDap6ZqYMOnXJjyurqkhLze3mXwHDkzFe5PUdI0m9LpqjhxQzJuvtWOTnQKTSkuJRCItfvbSQUVFRShypqnA6hbAeaMi1A5P/cnpjetWugpLztxMGJhZve//dFJpKQBL0/B7F5Z6EJacaSpldQtaX7uOyo5w/aScfQtI/Dt/sGGJw/cbFv+3pqaGvLy8fcMarz80nk+y88UgyyA7w8jKgOzM2OPsDMjOoNE5Z5H4T80HgyohUglFm2PdMYTj86v1mNQLS85kmrYiYHjC82HAlmamKTKzLKA3cMAxfe5+L3AvwNSpU33GjBktLvxjSQQ8FLGtKS0vP2jKmWIFBZSWloYia2je0/QUaN1qeYpDE5bPRFhyQqOsBQUAaZk9LO9pWHKmqZTVLTi02tUewvSZCEVWrcekXFhyJnNcyAJgnJkdbmbdgAuBeY2mmQdcFn/8WeCfqTi+WkTkEKluiUjYqG6JSLMsme+6mZ0F3EHsErR/cPefmdmPgYXuPs/McoGHgMnEtvhcuPdE2oPMcyewsa2/QBv0JwXHgHcA5Uy9sGQNOudIdx8Q4PLbRHUrUGHJCeHJqpzJUd1qer5B1q6gPxOtEZasyplaQedMqm4l1bR1Rma20N2nBp2jJcqZemHJGpac0nHC8pkIS04IT1bllLAK02ciLFmVM7XCklOXzRIREREREUljatpERERERETSWFdu2u4NOkCSlDP1wpI1LDml44TlMxGWnBCerMopYRWmz0RYsipnaoUiZ5c9p01ERERERCQMuvKeNhERERERkbSnpg0ws2+YmZtZ/6CzNMXMfmlm75rZMjP7k5kVBJ0pkZmdaWarzGytmX076DxNMbPhZvYvM1tpZm+b2Q1BZzoYM8s0syVm9lzQWSQ9qW61jepW6qluSUtUt9omDHULVLvaS5dv2sxsOPBR4P2gsxzES8AEdz8WWA3cHHCefcwsE7gb+ARwNHCRmR0dbKomNQBfd/ejgBOB69M05143ACuDDiHpSXWrbVS32o3qljRLdattQlS3QLWrXXT5pg24HbgJSNuT+9z9b+7eEH/6BjAsyDyNTAPWuvt77l4HPAacG3CmA7j7VndfHH9cTuzLOTTYVE0zs2HA2cB9QWeRtKW61TaqWymmuiVJUN1qm1DULVDtai9dumkzs3OAze7+VtBZWuFK4IWgQyQYCmxKeF5Emn4x9zKzUcBk4M1gkzTrDmJ/2KJBB5H0o7qVEqpbqae6Jc1S3UqJ0NUtUO1KpaygA7Q3M/s7cFgTo74LfAf4WMcmatrBcrr7M/Fpvktsl/PcjszWAmtiWNpuRTOzfOAp4EZ33xN0nsbM7JPADndfZGYzgs4jwVDdaneqWymkuiWgutUBQlW3QLUr1Tp90+buH2lquJlNBA4H3jIziO0CX2xm09x9WwdGBJrPuZeZXQZ8Epjp6XWfhiJgeMLzYcCWgLIclJllEysec9396aDzNONDwDlmdhaQC/Qys4fd/fMB55IOpLrV7lS3Ukt1S1S32l9o6haodrUH3actzsw2AFPdfVfQWRozszOB24APu/vOoPMkMrMsYifrzgQ2AwuAi9397UCDNWKxvxRzgGJ3vzHoPMmIb/X5hrt/Mugskp5Utw6N6lb7Ud2SlqhuHZqw1C1Q7WovXfqcthC5C+gJvGRmS83st0EH2it+wu5s4EViJ5o+kY4FhNjWlEuBM+Lv4dL4lhURaR+qW22nuiXSsVS3UkO1qx1oT5uIiIiIiEga0542ERERERGRNKamTUREREREJI2paRMREREREUljatpERERERETSmJo2ERERERGRNKamTUREREREJI2paRMREREREUljatpERERERETS2P8HtK0Z/u2+KscAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 1080x216 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-5, 5, 1000)\n",
    "plt.figure(figsize=(15,5))\n",
    "plt.subplot(2,3,1)\n",
    "plt.plot(x, uniform.pdf(x, 0, 4))\n",
    "plt.axvline(uniform.mean(0, 4), color='r')\n",
    "plt.title(r'Hustota $U(0, 4)$')\n",
    "plt.subplot(2,3,2)\n",
    "plt.plot(x[x>0], expon.pdf(x[x>0], scale=1/5))\n",
    "plt.axvline(expon.mean(scale=1/5), color='r')\n",
    "plt.title(r'Hustota $Exp(5)$')\n",
    "plt.subplot(2,3,3)\n",
    "plt.plot(x, norm.pdf(x, loc=0, scale=1))\n",
    "plt.axvline(norm.mean(0, 4), color='r')\n",
    "plt.title(r'Hustota $N(0,1)$')\n",
    "plt.figure(figsize=(15,3))\n",
    "plt.subplot(2,3,4)\n",
    "plt.plot(x, uniform.cdf(x, 0, 4))\n",
    "plt.axvline(uniform.mean(0, 4), color='r')\n",
    "plt.title(r'Distr. fce $U(0, 4)$')\n",
    "plt.subplot(2,3,5)\n",
    "plt.plot(x, expon.cdf(x, scale=1/5))\n",
    "plt.axvline(expon.mean(scale=1/5), color='r')\n",
    "plt.title(r'Distr. fce $Exp(5)$')\n",
    "plt.subplot(2,3,6)\n",
    "plt.plot(x, norm.cdf(x, loc=0, scale=1))\n",
    "plt.axvline(norm.mean(0, 4), color='r')\n",
    "plt.title(r'Distr. fce $N(0,1)$')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rovnoměrné rozdělení (spojité) $U(a,b)$\n",
    "\n",
    "[Rovnoměrné rozdělení](https://en.wikipedia.org/wiki/Uniform_distribution_(continuous%29) popisuje jevy, jejichž pravděpodobnost je rovnoměrně rozdělena na intervalu [a,b].\n",
    "\n",
    "### Hustota pravděpodobnosti\n",
    "$$\n",
    "f(x) = \\begin{cases}\\frac{1}{b-a} & \\text{pro}\\ x\\in [a,b], \\\\ 0 & \\text{jinde.}\\end{cases}\n",
    "$$\n",
    "\n",
    "### Momenty\n",
    "$$\n",
    "EX = \\frac{a+b}{2}, \\qquad \\operatorname{var}X = \\frac{(b-a)^2}{12}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "V následujícím kódu můžete zkusit měnit parametry $a, b$ a sledovat, co to dělá s hustotou a distribuční funkcí."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "a, b = (0, 3)   ### Zkuste měnit\n",
    "\n",
    "scale = np.abs(b-a)\n",
    "x = np.linspace(-5, 5, 1000)\n",
    "plt.figure(figsize=(15,3))\n",
    "plt.subplot(1,2,1)\n",
    "plt.plot(x, uniform.pdf(x, a, scale=scale))\n",
    "plt.axvline(uniform.mean(a, scale=scale), color='r')\n",
    "plt.title('Hustota U({0},{1}), stř. hod. EX={2:.2f}'.format(a, b, uniform.mean(a,scale=scale)))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x)')\n",
    "plt.subplot(1,2,2)\n",
    "plt.plot(x, uniform.cdf(x, a, scale=scale))\n",
    "plt.axvline(uniform.mean(a, scale=scale), color='r')\n",
    "plt.title('Distribuční funkce U({0},{1})'.format(a, b))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('F(x)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Aplikace\n",
    "Uvažujme, že máme odhadnout Ludolfovo číslo $\\pi=3.1415\\ldots$ náhodným vzorkováním (metodou Monte Carlo :). Vezmeme si k tomu pár faktů:\n",
    "\n",
    "1. Plocha kruhu: $S_\\circ = \\pi r^2$. Zvolíme-li $r=1$, zbyde $S_\\circ = \\pi$.\n",
    "2. Čtvrtina kruhu = vyšší efektivita vzorkování. $S_{\\frac{\\circ}{4}} = \\frac{\\pi}{4}$.\n",
    "3. Čtvrtkruh vepíšeme do čtverce o straně $r=1$, tedy o ploše $S_\\square = r^2 = 1$.\n",
    "4. Pravděpodobnost, že náhodný vzorek rovnoměrně padající do čtverce spadne i do kruhu je \n",
    "\n",
    "$$P(x_i \\in S_{\\frac{\\circ}{4}}) = \\frac{S_{\\frac{\\circ}{4}}}{S_\\square} = \\frac{\\pi}{4}.$$\n",
    "\n",
    "Z toho plyne, že $\\pi = 4 \\cdot P(x_i \\in S_{\\frac{\\circ}{4}})$. Navzorkujeme tedy z rovnoměrného rozdělení (resp. ze dvou rovnoměrných rozdělení $U[0,1]$ - pro každou osu) a spočítáme poměr vzorků uvnitř čtvrtkruhu ke všem vzorkům - ten odpovídá pravděpodobnosti $P(x_i \\in S_{\\frac{\\circ}{4}})$. Pak už jenom násobíme čtyřmi.\n",
    "\n",
    "<< sem přijde obrázek >>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Odhad pi: 3.14260\n"
     ]
    }
   ],
   "source": [
    "def odhad_pi(nsamples):\n",
    "    samples = np.random.uniform(low=0, high=1, size=(nsamples, 2))\n",
    "    samples_dist_from_origin = np.sqrt((samples**2).sum(axis=1))\n",
    "    samples_in_circle = np.sum(samples_dist_from_origin <= 1)\n",
    "    probability = samples_in_circle / nsamples\n",
    "    pi = 4 * probability\n",
    "    return pi\n",
    "\n",
    "nsamples = 100000\n",
    "print('Odhad pi: {0:.5f}'.format(odhad_pi(nsamples)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exponenciální rozdělení $\\mathcal{E}xp(\\lambda)$\n",
    "\n",
    "[Exponenciální rozdělení](https://en.wikipedia.org/wiki/Exponential_distribution) s parametrem $\\lambda>0$ (angl. rate, počet událostí za jednotku času) popisuje nezápornou náhodnou veličinu $X\\geq 0$. Používá se např. pro modelování času mezi dvěma událostmi s Poissonovým rozdělením (např. uvolnění částice při rozpadu atomu) či dotazy na server. Podobně, jako geometrické v diskrétním světě, je ve spojitém světě toto rozdělení **bezpaměťové**, tj. uvažujeme-li čas $T\\sim\\mathcal{E}xp(\\lambda)$, pak\n",
    "\n",
    "$$\n",
    "P(T>t+s|T>s) = P(T>t).\n",
    "$$\n",
    "\n",
    "(dokažte pomocí podmíněných pravděpodobností - je to jednoduché).\n",
    "\n",
    "### Hustota pravděpodobnosti\n",
    "$$\n",
    "f(x) = \\lambda e^{-\\lambda x}, \\qquad \\lambda>0.\n",
    "$$\n",
    "\n",
    "### Momenty\n",
    "$$\n",
    "EX = \\frac{1}{\\lambda}, \\qquad \\operatorname{var}X = \\frac{1}{\\lambda^2}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1080x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "lambd = 3   ### Zkuste měnit\n",
    "\n",
    "x = np.linspace(-5, 5, 1000)\n",
    "plt.figure(figsize=(15,3))\n",
    "plt.subplot(1,2,1)\n",
    "plt.plot(x, expon.pdf(x, scale=1/lambd))\n",
    "plt.axvline(expon.mean(scale=1/lambd), color='r')\n",
    "plt.title('Hustota Exp({0}), stř. hod. EX={1:.2f}'.format(lambd, expon.mean(scale=1/lambd)))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x)')\n",
    "plt.subplot(1,2,2)\n",
    "plt.plot(x, expon.cdf(x, scale=1/lambd))\n",
    "plt.axvline(expon.mean(scale=1/lambd), color='r')\n",
    "plt.title('Distr. fce Exp({0}), stř. hod. EX={1:.2f}'.format(lambd, expon.mean(scale=1/lambd)))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('F(x)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Příklad\n",
    "\n",
    "Máme klastr redundantních serverů zpracovávajících elektronické platby. Víme, že mimo špičku chodí na jednotlivé servery v průměru 10 transakcí za sekundu. Označíme čas $T$ mezi jednotlivými transakcemi. Určete a **interpretujte**:\n",
    "\n",
    "1. $\\lambda$\n",
    "2. Střední hodnotu a rozptyl.\n",
    "3. $P(T<\\frac{1}{10}s)$.\n",
    "4. $P(T>\\frac{1}{10}s)$.\n",
    "5. $P\\left(\\frac{1}{10}<T<\\frac{2}{10}\\right)$\n",
    "6. $P\\left(T>\\frac{5}{10}\\big|T>\\frac{1}{10}\\right)$\n",
    "\n",
    "Odpovědi:\n",
    "\n",
    "Vidíme $P(T<t)$ a $P(T>t)$, budeme tedy potřebovat distribuční funkci a funkci přežití. Tak si je spočteme :)\n",
    "\n",
    "Distribuční funkce:\n",
    "\n",
    "$$\n",
    "F(t) = \\int_{-\\infty}^t f(u)du =\n",
    "\\begin{cases}\n",
    "0 &\\text{pro}\\ t<0,\\\\\n",
    "\\int_{0}^{t} f(u)du = \\int_0^t \\lambda e^{-\\lambda u}du = \\left[ e^{-\\lambda u}\\right]_0^t = 1-e^{-\\lambda t} &\\text{pro}\\ x\\geq 0.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Funkce přežití:\n",
    "\n",
    "$$\n",
    "S(t) = P(T>t) =1-F(t) = e^{-\\lambda t}.\n",
    "$$\n",
    "\n",
    "**[Výsledky níže vznikly v presu, prosím o report chyb]**\n",
    "1. $\\lambda = 10$\n",
    "2. $EX = \\frac{1}{10}$ a $\\operatorname{var}X = \\frac{1}{100}$.\n",
    "3. $F\\left(\\frac{1}{10}\\right) = 1-e^{-1}$\n",
    "4. $S\\left(\\frac{1}{10}\\right) = e^{-1}$\n",
    "5. $F\\left(\\frac{2}{10}\\right) - F\\left(\\frac{1}{10}\\right) = e^{-1}-e^{-2}$.\n",
    "6. \n",
    "\n",
    "$$\n",
    "P\\left(T>\\frac{5}{10}\\bigg|T>\\frac{1}{10}\\right)\n",
    "= \\frac{P\\left(\\left\\{T>\\frac{5}{10}\\right\\} \\cap \\left\\{T>\\frac{1}{10}\\right\\}\\right)}\n",
    "{P\\left(T>\\frac{1}{10}\\right)}\n",
    "=\\frac{P\\left(T>\\frac{5}{10}\\right)}\n",
    "{P\\left(T>\\frac{1}{10}\\right)}\n",
    "= \\frac{e^{-5}}{e^{-1}} = e^{-4}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normální rozdělení $\\mathcal{N}(\\mu, \\sigma^2)$\n",
    "\n",
    "[Normální - též Gaussovo či gaussovské - rozdělení](https://en.wikipedia.org/wiki/Normal_distribution) je jedno z nejpopulárnějších vůbec. Má dva parametry - **střední hodnotu** $\\mu\\in\\mathbb{R}$ a **rozptyl** $\\sigma^2$, které jsou zároveň $EX$  a $\\operatorname{var}X$. Náhodná veličina $X\\in\\mathbb{R}$.\n",
    "\n",
    "Odmocnina rozptylu - $\\sigma$ - se nazývá **směrodatnou odchylkou**. Tu známe ze zápisu kde je $\\pm$, například v $(3\\pm 0.001) \\text{kg}$ je řečeno, že něco váží 3kg a zpravidla 3 směrodatné odchylky jsou 0.001kg. Časem se dozvíme, že 3 směrodatné odchylky na každou stranu od střední hodnoty pokrývají naprostou většinu hodnot.\n",
    "\n",
    "Normální rozdělení $\\mathcal{N}(0, 1)$ se nazývá **standardní normální rozdělení**. Veličiny se často transformují tak, aby měly stejné momenty a porovnávají s tímto rozdělením - normalita je prostě populární :) Jednou z obrovských výhod normálního rozdělení je, že se s ním velmi dobře počítá.\n",
    "\n",
    "\n",
    "### Hustota pravděpodobnosti\n",
    "$$\n",
    "f(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{-\\frac{1}{2\\sigma^2}(x-\\mu)^2}, \\qquad \\mu\\in\\mathbb{R}, \\sigma^2>0.\n",
    "$$\n",
    "\n",
    "### Momenty\n",
    "$$\n",
    "EX = \\mu, \\qquad \\operatorname{var}X = \\sigma^2.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "mi = 0    ### Zkuste měnit\n",
    "sigma = 1 ### Zkuste měnit\n",
    "\n",
    "x = np.linspace(-10, 10, 1000)\n",
    "plt.figure(figsize=(15,3))\n",
    "plt.subplot(1,2,1)\n",
    "plt.plot(x, norm.pdf(x, loc=mi, scale=sigma))\n",
    "plt.axvline(mi, color='r')\n",
    "plt.title('Hustota N({0},{1}), stř. hod. EX={2:.2f}'.format(mi, sigma**2, mi))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('f(x)')\n",
    "plt.subplot(1,2,2)\n",
    "plt.plot(x, norm.cdf(x, loc=mi, scale=sigma))\n",
    "plt.axvline(mi, color='r')\n",
    "plt.title('Distr. fce N({0},{1}), stř. hod. EX={2:.2f}'.format(mi, sigma**2, mi))\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('F(x)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Normální rozdělení vystihuje mnoho reálných dat, samozřejmě s přihlédnutím na jistá zjednodušení. Např. IQ se uvažuje rozdělené $\\mathcal{N}(100, 15^2)$. Jelikož ale normální rozdělení má nosič hustoty (tj. množinu, kde je $f(x)>0$) celou reálnou osu, platí, že pokud by IQ bylo přesně normální, musí být nenulová i pravděpodobnost určitých extrémů. Zkusme:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P(IQ<0) = 1.31e-11\n",
      "P(IQ<30) = 1.53e-06\n",
      "P(IQ<50) = 4.29e-04\n",
      "P(IQ>180) = 4.82e-08\n"
     ]
    }
   ],
   "source": [
    "mi = 100\n",
    "sigma = 15\n",
    "print('P(IQ<0) = {0:.2e}'.format(norm.cdf(0, loc=100, scale=15)))\n",
    "print('P(IQ<30) = {0:.2e}'.format(norm.cdf(30, loc=100, scale=15)))\n",
    "print('P(IQ<50) = {0:.2e}'.format(norm.cdf(50, loc=100, scale=15)))\n",
    "print('P(IQ>180) = {0:.2e}'.format(1-norm.cdf(180, loc=100, scale=15)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Teoreticky bychom tedy mohli potkat člověka se záporným IQ. Tyto nedokonalosti lze řešit ořezáním konců a renormalizací, ovšem takové rozdělení je z hlediska dalšího počítání peklo. Tak se raději smíříme s nedokonalostmi modelu :)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Přehled uvedených spojitých rozdělení\n",
    "\n",
    "| Rozdělení | Parametry | $f(x)$ | $\\operatorname{E}X$ | $\\operatorname{var}X$ | `scipy` |\n",
    "|---|---|---|---|---|---|\n",
    "|[Rovnoměrné](https://en.wikipedia.org/wiki/Uniform_distribution_&#040;continuous&#041;) | $a,b\\in\\mathbb{R}$ | $\\begin{cases}\\frac{1}{b-a} & \\text{pro}\\ x\\in [a,b], \\\\ 0 & \\text{jinde.}\\end{cases}$ | $\\frac{a+b}{2}$ | $\\frac{(b-a)^2}{12}$ | [`uniform`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.uniform.html) |\n",
    "|[Exponenciální](https://en.wikipedia.org/wiki/Exponential_distribution) | $\\lambda>0$ | $\\lambda e^{-\\lambda x}$ | $\\frac{1}{\\lambda}$ | $\\frac{1}{\\lambda^2}$ | [`expon`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.expon.html)|\n",
    "|[Normální](https://en.wikipedia.org/wiki/Normal_distribution)| $\\mu\\in\\mathbb{R},\\sigma^2>0$ | $\\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{-\\frac{1}{2\\sigma^2}(x-\\mu)^2}$ | $\\mu$ | $\\sigma^2$ | [`norm`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html)|\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Příklady ze slajdů BI-PST"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.1\n",
    "**Uvažujme funkci**\n",
    "\n",
    "$$\n",
    "F_X(x) =\n",
    "\\begin{cases}\n",
    "\\frac{x^2}{2} &\\text{pro}\\ x\\in[0,1), \\\\\n",
    "2x - \\frac{x^2}{2} - 1 &\\text{pro}\\ x\\in[1,2],\\\\\n",
    "0 &\\text{jinde}.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**a) Je funkce $F_X(x)$ distribuční funkcí? Pokud ne, je možné ji částečně předefinovat tak, aby byla? V kladném případě uvažujte do dalších bodů tuto předefinovanou variantu.**\n",
    "\n",
    "(nakreslete si obrázek) Připomeňme, že vlastnosti d.f. jsou (i) monotónie - funkce je neklesající a (ii) $\\lim_{x\\to \\infty} F_X(x) = 1$. Je tedy potřeba ji předefinovat:\n",
    "\n",
    "$$\n",
    "F_X(x) =\n",
    "\\begin{cases}\n",
    "0 &\\text{pro}\\ x<0,\\\\\n",
    "\\frac{x^2}{2} &\\text{pro}\\ x\\in[0,1), \\\\\n",
    "2x - \\frac{x^2}{2} - 1 &\\text{pro}\\ x\\in[1,2],\\\\\n",
    "1 &\\text{pro}\\ x>2.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**b) Jedná se o spojitou náhodnou veličinu?**\n",
    "\n",
    "Ano, viz přednáška. Distribuční funkce je všude spojitá, $\\lim_{x\\to c^-} F(x) = \\lim_{x\\to c^+} F(x)$ pro všechna $c\\in D(f)$ kde $D$ je definiční obor.\n",
    "\n",
    "**c) Najděte její hustotu.**\n",
    "\n",
    "Jednoduše derivací zjistíme, že\n",
    "\n",
    "$$\n",
    "f_X(x) =\n",
    "\\begin{cases}\n",
    "0 &\\text{pro}\\ x<0,\\\\\n",
    "x &\\text{pro}\\ x\\in[0,1), \\\\\n",
    "2 - x &\\text{pro}\\ x\\in[1,2],\\\\\n",
    "0 &\\text{pro}\\ x>2.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**d) Najděte pravděpodobnost $P\\left(X\\in\\left[\\frac{1}{2}, \\frac{3}{2}\\right]\\right)$.**\n",
    "\n",
    "$$\n",
    "P\\left(X\\in\\left[\\frac{1}{2}, \\frac{3}{2}\\right]\\right) = F\\left(\\frac{3}{2}\\right) - F\\left(\\frac{1}{2}\\right) = \\frac{7}{8} - \\frac{1}{8} = \\frac{3}{4}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.2\n",
    "\n",
    "**Nechť $X$ je spojitá náhodná veličina s hustotou**\n",
    "![5.2](img/52.jpg)\n",
    "\n",
    "**a) Najděte hodnotu konstanty $h$.**\n",
    "\n",
    "Víme, že hustota pravděpodobnosti se musí integrovat do jedné. Tedy\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^{\\infty} f(x) dx =\n",
    "\\underbrace{\\frac{1}{2} \\cdot 1 \\cdot h}_{\\text{trojúhelník}} + \n",
    "\\underbrace{1 \\cdot h}_{\\text{obdélník}} = 1 \\Rightarrow h=\\frac{2}{3}.\n",
    "$$\n",
    "\n",
    "**b) Najděte analytické vyjádření hustoty $f(x)$.**\n",
    "\n",
    "$$\n",
    "f(x) =\n",
    "\\begin{cases}\n",
    "\\frac{2}{3}x &\\text{pro}\\ x\\in[0,1),\\\\\n",
    "\\frac{2}{3}  &\\text{pro}\\ x\\in[1,2],\\\\\n",
    "0 &\\text{jinde}.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**c) Spočtěte střední hodnotu $EX$ a rozptyl $\\operatorname{var}X$.**\n",
    "\n",
    "$$\n",
    "EX = \\int x f(x) dx = \\int_0^1 x\\cdot \\frac{2}{3}x dx + \\int_1^2 x\\cdot \\frac{2}{3} dx = \\frac{11}{9}.\n",
    "$$\n",
    "\n",
    "Protože $\\operatorname{var}X = EX^2 - (EX)^2$, spočteme si ještě jednu střední hodnotu,\n",
    "\n",
    "$$\n",
    "EX^2 = \\int x^2 f(x)dx = \\int_0^1 x^2\\cdot \\frac{2}{3}x dx + \\int_1^2 x^2\\cdot \\frac{2}{3} dx = \\frac{31}{18}.\n",
    "$$\n",
    "\n",
    "Tedy\n",
    "\n",
    "$$\n",
    "\\operatorname{var}X = \\frac{31}{18} - \\left(\\frac{11}{9}\\right)^2 = \\frac{37}{162}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.3\n",
    "\n",
    "Mějme funkci $g:\\mathbb{R}\\to\\mathbb{R}$ zadanou následovně:\n",
    "\n",
    "$$\n",
    "g(x) =\n",
    "\\begin{cases}\n",
    "2x &\\text{pro}\\ x\\in(0,1],\\\\\n",
    "4-2x &\\text{pro}\\ x\\in(1,2],\\\\\n",
    "0 &\\text{pro}\\ x\\in(-\\infty, 0]\\cup (2, \\infty).\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "**a) Najděte takovou hodnotu $a\\in\\mathbb{R}$, aby funkce $f(x) = a\\cdot g(x)$ byla hustotou pravděpodobnosti nějaké náhodné veličiny.**\n",
    "\n",
    "Opět budeme hledat $a$ takové, aby integrál $f(x)$ byl roven jedné:\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty a\\cdot g(x)dx = \n",
    "a \\left[ \\int_1^2 4-2x dx + \\int_0^1 2x dx\\right] = 1 \\Longrightarrow a=\\frac{1}{2}.\n",
    "$$\n",
    "\n",
    "**b) Najděte střední hodnotu a rozptyl náhodné veličiny s hustotou pravděpodobnosti $f(x)$.**\n",
    "\n",
    "Budeme postupovat stejně jako v předchozím příkladu, spočteme $EX$ a $EX^2$:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "EX &= \\int_{-\\infty}^\\infty x f(x) dx = \\int_0^1 x\\cdot x dx + \\int_1^2 x(2-x)dx = 1,\\\\\n",
    "EX^2 &= \\int_{-\\infty}^\\infty x^2 f(x) dx = \\int_0^1 x^2\\cdot x dx + \\int_1^2 x^2(2-x)dx = \\frac{7}{6},\\\\\n",
    "\\operatorname{var}X &= EX^2 - (EX)^2 = \\frac{7}{6} -1 = \\frac{1}{6}.\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 5.4\n",
    "\n",
    "**Máme vypozorováno, že průměrná délka intervalu mezi příjezdy tramvaje č. 20 na zastávku Vítězné náměstí je ve špičce 5 minut. Označme $T$ náhodnou veličinu, která měří délku intervalu mezi dvěma příjezdy tramvaje.**\n",
    "\n",
    "**a) Modelujte $T$ pomocí rovnoměrného, exponenciálního a normálního rozdělení tak, aby $ET = 5$min.**\n",
    "\n",
    "Musíme nastavit parametry jednotlivých rozdělení tak, aby byla střední hodnota 5:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "T &\\sim \\mathcal{U}(0, 10), \\\\\n",
    "T &\\sim \\mathcal{E}xp\\left(\\frac{1}{5}\\right),\\\\\n",
    "T &\\sim \\mathcal{N}(5, \\sigma^2).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "**b) Jaký je rozptyl $T$ pro jednotlivá rozdělení?**\n",
    "\n",
    "Jednoduchým dosazením do vzorečků pro rozptyl postupně dostaneme $\\frac{25}{3}, 25$ a $\\sigma^2$.\n",
    "\n",
    "**c) Jaká je pravděpodobnost, že interval bude delší než 6 minut, tj. $P(T>6)$?**\n",
    "\n",
    "Nejprve obecně, víme, že $F(X) = P(X\\leq x)$ a tedy $P(X>x) = 1-F(x) = S(x)$, což je tzv. funkce přežití (survival function). Určíme je pro rovnoměrné a exponenciální rozdělení, normální nás čeká v 6. cvičení.\n",
    "\n",
    "*Rovnoměrné rozdělení $\\mathcal{U}(a,b)$:*\n",
    "\n",
    "$$\n",
    "F(x) =\n",
    "\\begin{cases}\n",
    "0 &\\text{pro}\\ x<a,\\\\\n",
    "\\int_{a}^x \\frac{1}{b-a}dt = \\frac{x-a}{b-a} &\\text{pro}\\ x\\in[a,b],\\\\\n",
    "1 &\\text{pro}\\ x>b.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Tedy $P(T>6) = 1-F(6) = 1-\\frac{6}{10} = 0.4$.\n",
    "\n",
    "*Exponenciální rozdělení $\\mathcal{E}xp(\\lambda)$:*\n",
    "\n",
    "$$\n",
    "F(x) = \\int_{-\\infty}^{x} = \\int_{0}^x \\lambda e^{-\\lambda t} dt = \\left[-e^{-\\lambda t}\\right]_0^x = 1 - e^{-\\lambda x}.\n",
    "$$\n",
    "\n",
    "Tedy $P(T>6) = 1-F(6) = e^{-\\frac{6}{5}}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.5\n",
    "\n",
    "**Uvažujme náhodnou veličinu $T\\sim \\mathcal{E}xp(\\lambda)$ z předchozího příkladu.**\n",
    "\n",
    "**a) Nalezněte předpis pro distribuční funkci a funkci přežití $T$.**\n",
    "\n",
    "Distribuční funkce (budu značit $X$ místo $T$):\n",
    "\n",
    "$$\n",
    "F(x) = \\int_{-\\infty}^x f(x)dx =\n",
    "\\begin{cases}\n",
    "0 &\\text{pro}\\ x<0,\\\\\n",
    "\\int_{0}^{x} f(x)dx = \\int_0^x \\lambda e^{-\\lambda t}dt = \\left[ e^{-\\lambda t}\\right]_0^x = 1-e^{-\\lambda x} &\\text{pro}\\ x\\geq 0.\n",
    "\\end{cases}\n",
    "$$\n",
    "\n",
    "Funkce přežití:\n",
    "\n",
    "$$\n",
    "S(x) = P(X>x) =1-F(x) = e^{-\\lambda x}.\n",
    "$$\n",
    "\n",
    "**b) Spočtěte pravděpodobnost, že délka intervalu mezi příjezdy bude rovna 5 min., tj. $P(T=5)$.**\n",
    "\n",
    "Pravděpodobnost spojité n.v. v bodě je rovna 0 (jaká je pravděpodobnost, že to bude s nekonečnou přesností právě 5 minut? ;))\n",
    "\n",
    "**c) Spočtěte pravděpodobnost, že délka intervalu mezi příjezdy bude větší než 5 minut, tj. $P(T>5)$.**\n",
    "\n",
    "Použitím funkce přežití dostaneme $P(T>5) = e^{-\\frac{5}{5}} = e^{-1}$.\n",
    "\n",
    "**d) Jaká je pravděpodobnost, že interval bude delší než 6 minut za podmínky, že čekáme už déle než 5 minut, tj. $P(T>6|T>5)$?**\n",
    "\n",
    "Podmíněná pravděpodobnost:\n",
    "\n",
    "$$\n",
    "P(T>6|T>5) = \\frac{P\\left( \\{T>6\\}\\cap \\{T>5\\} \\right)}{P(T>5)}\n",
    "=\\frac{P(T>6)}{P(T>5)} = \\frac{e^{-\\frac{6}{5}}}{e^{-\\frac{5}{5}}} = e^{-\\frac{1}{5}}.\n",
    "$$\n",
    "\n",
    "**e) Jaká je pravděpodobnost, že interval bude nejvýše 6 minut za podmínky, že už čekáme déle než 5 minut, tj. $P(T\\leq 6|T>5)$?**\n",
    "\n",
    "Podobně jako v předchozím případě,\n",
    "\n",
    "$$\n",
    "P(T\\leq 6|T>5) = \\frac{P\\left( \\{T\\leq 6\\}\\cap \\{T>5\\} \\right)}{P(T>5)}=\n",
    "\\frac{P(T\\in(5,6])}{P(T>5)} = \\frac{F(6) - F(5)}{S(5)} = 1-e^{-\\frac{1}{5}}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5.6\n",
    "\n",
    "**Ukažte bezpaměťovost exponenciálního rozdělení, tj. že pro $T\\sim\\mathcal{E}xp(\\lambda)$ platí**\n",
    "\n",
    "$$\n",
    "P(T>t+s|T>s) = P(T>t).\n",
    "$$\n",
    "\n",
    "To uděláme snadno podobně, jako v předchozím případě v posledních dvou bodech. Prostě rozepíšeme podmíněné pravděpodobnosti, nalezneme průnik a ono to vypadne samo :) Zkuste!"
   ]
  }
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