06-integration.Rmd

# Integration {#integ}

This chapter deals with abstract and Monte Carlo integration.

The students are expected to acquire the following knowledge:

**Theoretical**

- How to calculate Lebesgue integrals for non-simple functions.

**R**

- Monte Carlo integration.


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```{r, echo = FALSE, warning = FALSE, message = FALSE}
togs <- T
library(ggplot2)
library(dplyr)
library(reshape2)
library(tidyr)
# togs <- FALSE
```

## Monte Carlo integration
```{exercise}
<span style="color:blue">Let $X$ and $Y$ be continuous random variables on the unit interval and $p(x,y) = 6(x - y)^2$. Use Monte Carlo integration to estimate the probability $P(0.2 \leq X  \leq 0.5, \: 0.1 \leq Y \leq 0.2)$. Can you find the exact value?</span>
```

<div class="fold">
```{r, echo = togs, message = FALSE, warning=FALSE, eval = togs}
set.seed(1)
nsamps <- 1000
V  <- (0.5 - 0.2) * (0.2 - 0.1)
x1 <- runif(nsamps, 0.2, 0.5)
x2 <- runif(nsamps, 0.1, 0.2)
f_1 <- function (x, y) {
  return (6 * (x - y)^2)
}
mcint  <- V * (1 / nsamps) * sum(f_1(x1, x2))
sdm    <- sqrt((V^2 / nsamps) * var(f_1(x1, x2)))

mcint
sdm

F_1 <- function (x, y) {
  return (2 * x^3 * y - 3 * x^2 * y^2 + 2 * x * y^3)
}
F_1(0.5, 0.2) - F_1(0.2, 0.2) - F_1(0.5, 0.1) + F_1(0.2, 0.1)
```
</div>

## Lebesgue integrals
```{exercise, name = "borrowed from Jagannathan"}
Find the Lebesgue integral of the following functions on ($\mathbb{R}$, $\mathcal{B}(\mathbb{R})$, $\lambda$).

a.
\begin{align}
  f(\omega) = 
    \begin{cases}
      \omega, & \text{for } \omega = 0,1,...,n \\
      0, & \text{elsewhere}
    \end{cases}
\end{align}
  
b.
\begin{align}
  f(\omega) = 
    \begin{cases}
      1, & \text{for } \omega = \mathbb{Q}^c \cap [0,1] \\
      0, & \text{elsewhere}
    \end{cases}
\end{align}
  
c.
\begin{align}
  f(\omega) = 
    \begin{cases}
      n, & \text{for } \omega = \mathbb{Q}^c \cap [0,n] \\
      0, & \text{elsewhere}
    \end{cases}
\end{align}
```

<div class="fold">
```{solution, echo = togs}


a.
\begin{align}
  \int f(\omega) d\lambda = \sum_{\omega = 0}^n \omega \lambda(\omega) = 0.
\end{align}

b.
\begin{align}
  \int f(\omega) d\lambda = 1 \times \lambda(\mathbb{Q}^c \cap [0,1]) = 1.
\end{align}

c.
\begin{align}
  \int f(\omega) d\lambda = n \times \lambda(\mathbb{Q}^c \cap [0,n]) = n^2.
\end{align}
```
</div>


```{exercise, name = "borrowed from Jagannathan"}
Let $c \in \mathbb{R}$ be fixed and ($\mathbb{R}$, $\mathcal{B}(\mathbb{R})$) a measurable space. If for any Borel set $A$, $\delta_c (A) = 1$ if $c \in A$, and $\delta_c (A) = 0$ otherwise, then $\delta_c$ is called a _Dirac measure_. Let $g$ be a non-negative, measurable function. Show that $\int g d \delta_c = g(c)$.
```

<div class="fold">
```{solution, echo = togs}
\begin{align}
  \int g d \delta_c &= \sup_{q \in S(g)} \int q d \delta_c \\
                    &= \sup_{q \in S(g)} \sum_{i = 1}^n a_i \delta_c(A_i) \\
                    &= \sup_{q \in S(g)} \sum_{i = 1}^n a_i \text{I}_{A_i}(c) \\
                    &= \sup_{q \in S(g)} q(c) \\
                    &= g(c)
\end{align}
```
</div>