Monte_Carlo_Approximate_Intergral_And_Expectation.qmd

---
title: MonteCarlo Excercise
format: gfm
editor: visual
---


## Problem: 1


Obtain an approximation for the expression below using simulation. 

$$
\begin{aligned}
& U \sim \mathcal{U}(0, 1), \\
& Y = \exp(U) \implies \\
& \text{Cov}(U, Y) = ?
\end{aligned}
$$



#### Theoretical solution


$$
\begin{aligned}
& \text{Cov}(U, Y) = \mathcal{E}(U \times Y) - \mathcal{E}(U \times Y) = \\
& \mathcal{E}(U \times \exp(U)) - \mathcal{E}(U)\times \mathcal{E}(\exp(U)) \\
& \mathcal{E}(U \times \exp(U)) = \int_0^1 u \times \exp(u) du = u\exp(u) - \int_0^1 \exp(u) du  = \\
& \left[u\exp(u) - \exp(u)\right]_0^1  = \exp(1) - \exp(1) - (0 - 1) = 1, \\
& \mathcal{E}(U) = \frac{1}{2}, \quad \mathcal{E}(\exp(U)) = \int_0^1 \exp(u)du = [\exp(u)]_0^1 = \exp(1) - \exp(0) = \\
& \exp(1) - 1 \implies \\
& \text{Cov}(U, Y) = 1 - \frac{1}{2} \times (\exp(1) - 1) = \frac{3 - \exp(1)}{2} \approx `r round((3 - exp(1))/2, 4)`
\end{aligned}
$$

#### Code: 


```{r}
#| warning: false
#| message: false
#| fig-width: 9
#| fig-height: 9

sim_fun_1 <- function(n){
    U = runif(n)
    Y = exp(U)
    covv = mean(U * Y) - mean(U) * mean(Y)
    return (covv)
}

num = c(1e+4, 1e+5, 1e+6, 1e+7)

result = unlist(lapply(num, sim_fun_1))

```


```{r}
#| eval: false

print(cbind(num, result))
```


```{r}
#| echo: false

data.frame(number_of_Iteration = num, result = result, 
                Theory_vlaue = round((3 - exp(1))/2, 6)) |> 
knitr :: kable(, align = "c", caption = 
                "Table of simulation Results")

```



***
***
***


## Problem 2: 


$$
\begin{aligned}
& U \sim \mathcal{U}(0, 1), \quad Y = \lfloor 10 \times U \rfloor + 1 \\
& \text{Cov}(U Y) = ?
\end{aligned}
$$



#### Theory Soloution

$$
\begin{aligned}
& Y = \begin{cases} 
y = 1 \iff U < \frac{1}{10}, \\
y = 2 \iff \frac{1}{10} U < \frac{2}{10} , \\
y = 3 \iff \frac{2}{10} U < \frac{3}{10}, \\
y = 4 \iff \frac{3}{10} U < \frac{4}{10}, \\
y = 5 \iff \frac{4}{10} U < \frac{5}{10}, \\
y = 6 \iff \frac{5}{10} U < \frac{6}{10}, \\
y = 7 \iff \frac{6}{10} U < \frac{7}{10}, \\
y = 8 \iff \frac{7}{10} U < \frac{8}{10}, \\
y = 9 \iff \frac{8}{10} U < \frac{9}{10}, \\
y = 10 \iff \frac{9}{10} U < \frac{10}{10}.
\end{cases} \implies \\
& \mathbb{P}(Y = y) = \begin{cases} 
y = 1 & \mathbb{P}(U < \frac{1}{10}) = \frac{1}{10} , \\
y = 2 &  \mathbb{P}(\frac{1}{10} < U < \frac{2}{10}) = \frac{1}{10}, \\
y = 3 &  \mathbb{P}(\frac{2}{10} < U < \frac{3}{10}) = \frac{1}{10}, \\
y = 4 &  \mathbb{P}(\frac{3}{10} < U < \frac{4}{10}) = \frac{1}{10}, \\
y = 5 &  \mathbb{P}(\frac{4}{10} < U < \frac{5}{10}) = \frac{1}{10}, \\
y = 6 &  \mathbb{P}(\frac{5}{10} < U < \frac{6}{10}) = \frac{1}{10}, \\
y = 7 &  \mathbb{P}(\frac{6}{10} < U < \frac{7}{10}) = \frac{1}{10}, \\
y = 8 &  \mathbb{P}(\frac{7}{10} < U < \frac{8}{10}) = \frac{1}{10}, \\
y = 9 &  \mathbb{P}(\frac{8}{10} < U < \frac{9}{10}) = \frac{1}{10}, \\
y = 10 & \mathbb{P}(\frac{9}{10} < U < \frac{10}{10}) = \frac{1}{10},
\end{cases}
\end{aligned}
$$

$$
\begin{aligned}
& \text{Cov}(Y \exp(Y)) = \mathcal{E}(Y \exp(Y)) - \mathcal{E}(Y) \times \mathcal{E}(\exp(Y)), \\
& \mathcal{E}(\exp(Y)) = \sum_{i = 1}^{10} \exp(i) \times \frac{1}{10} = \frac{1}{10}\times \sum_{i = 1}^{10} \exp(i) \approx `r round(1/10 * sum(exp(1:10)), 6)`, \\
& \mathcal{E}{Y} = \frac{1}{10}\sum_{i = 1}^{10} i = \frac{1}{10}\frac{10 \times 11}{2} = \frac{11}{2}, \\
& \mathcal{E}(Y \exp(Y)) = \frac{1}{10}\sum_{i = 1}{10} i \times \exp(i) \approx `r round(1/10 * sum(((1:10) * exp(1:10))), 6)`\\
& \text{Cov}(Y, \exp(Y)) = `r round(1/10 * sum(((1:10) * exp(1:10))), 6)` - \frac{11}{2} \times `r round(1/10 * sum(exp(1:10)), 6)` \approx `r round((1/10 * sum(((1:10) * exp(1:10)))) - (11 / 2) * (1/10 * sum(exp(1:10))), 6)`
\end{aligned}
$$


#### Simulate With R 



```{r}
#| warning: false
#| message: false
#| fig-width: 9
#| fig-height: 9

sim_fun_2 <- function(n){
    U = runif(n)
    Y = floor(10 * U) + 1
    Y2= exp(Y)
    covv = mean(Y * Y2) - mean(Y) * mean(Y2)
    return (covv)
}

num = c(1e+4, 1e+5, 1e+6, 1e+7)

result = unlist(lapply(num, sim_fun_2))

```


```{r}
#| eval: false

print(cbind(num, result))
```


```{r}
#| echo: false

temp <- round((1/10 * sum(((1:10) * exp(1:10)))) - (11 / 2) * (1/10 * sum(exp(1:10))), 6)
data.frame(number_of_Iteration = num, result = result, 
                Theory_vlaue = temp) |> 
knitr :: kable(, align = "c", caption = 
                "Table of simulation Results")

```


***
***
***


## Problem iii

$$
\begin{aligned} 
a: \quad & \int_0^1 \exp(\exp(x)) ~dx, \\
~~ & \\
b: \quad & \int_{0}^{1} (1 - x^2)^\frac{3}{2}~dx, \\
~~ & \\
c: \quad & \int_{-2}^{+2} \exp(x + x^2)~dx, \\
~~ & \\
d: \quad & \int_{-\infty}^{+\infty} \exp(-x^2)~dx
\end{aligned}
$$


> Soloution a: 

```{r}
f <- function(x) exp(exp(x))

Real_value <- integrate(f, 0, 1)$value

## Simulation Approach 

n <- 1e+5
U <- runif(n)
simulation_value <- mean(f(U))

c(Real_value = Real_value, Simulation_value = simulation_value)
```


***


> soloution b: 


```{r}
f <- function(x) (1 - x**2)**(3/2)

Real_value <- integrate(f, 0, 1)$value

## Simulation Approach 

n <- 1e+5
U <- runif(n)
simulation_value <- mean(f(U))

c(Real_value = Real_value, Simulation_value = simulation_value)
```


***

> soloution c: 


```{r}
f <- function(x) exp(x + x**2)

Real_value <- integrate(f, -2, 2)$value

## Simulation Approach 

n <- 1e+5
U <- runif(n, min = -2, max = 2)
simulation_value <- 4 * mean(f(U))

c(Real_value = Real_value, Simulation_value = simulation_value)

```




***


> soloution d:



$$
\begin{aligned} 
& \int_{-\infty}^{+\infty} \exp(-x^2)~dx \overset{\text{even function}}{=} 2\int_{0}^{+\infty} \exp(-x^2)~dx, \\
& ~~ \\
& \text{if} ~~ u = \frac{1}{x^2 + 1} \implies x = \sqrt{\frac{1}{u} - 1} \\ 
& ~~\\
& \text{and}~~ du = dx \times \frac{-2x}{(x^2 + 1)^2} \implies du \times \frac{(x^2 + 1)^2}{-2x} = dx\\
&~~\\
& \int_{-\infty}^{+\infty} \exp(-x^2)~dx = 2 \times \int_1^0 \frac{1}{(-2 \times\sqrt{\frac{1}{u} - 1})} \times \frac{1}{u^2} \times \exp(\frac{u - 1}{u})~du \\
& ~~\\
& = \int_0^1 \frac{1}{(2 \times\sqrt{\frac{1}{u} - 1})} \times \frac{1}{u^2} \times \exp(\frac{u - 1}{u})~du \\
& ~~\\
\end{aligned}
$$



#### Exact answer

$$
\begin{aligned}
& \text{if}~~ X \sim \mathcal{N}\left(0,~ \frac{\sqrt{2}}{2}\right) \implies \\
& f_X(x) = \frac{1}{\sqrt{2 \times \pi \times \left(\frac{\sqrt{2}}{2}\right)^2}} \exp\left(-\frac{1}{2 \left(\frac{\sqrt{2}}{2}\right)^2}x^2\right) \\
& = \frac{1}{\sqrt{\pi}}\times \exp(-x^2) \implies \\
& \int_{-\infty}^{+\infty} \exp(-x^2)~dx = \sqrt{\pi} \int_{-\infty}^{+\infty} \frac{1}{\sqrt{\pi}}\times \exp(-x^2)~dx = \sqrt{\pi}
\end{aligned}
$$



>> Simulation Code: 

```{r}
f <- function(u) {
    temp1 <- 1 / (2 * sqrt(1/u - 1))
    temp2 <- 1/u**2 * exp((u - 1)/u)
    temp3 <- 2 * temp1 * temp2
    return(temp3)
}



R_Integrate_Estimate <- integrate(f, 0, 1)$value
Exact_Result <- sqrt(pi)

## Simulation Approach 

n <- 1e+5
U <- runif(n)
simulation_value <- mean(f(U))

c(R_Integrate_Estimate = R_Integrate_Estimate, Exact_Soloution = Exact_Result, Simulation_value = simulation_value)

```