# Exercise 13 : Gaussian integral and Variance reduction

```{r}
set.seed(123)
n <- 10000

# Normal distribution
h1 <- function(x) {
  sqrt(2 * pi) *
    exp(-x^2 / 2) *
    (x <= 2) *
    (x >= 0)
}

# Uniform (0, 2) distribution
h2 <- function(x) {
  2 * exp(-x^2)
}

X1 <- rnorm(n)
X2 <- runif(n, 0, 2)

cat(sprintf("Integral of h1(x) using normal distribution is %f and variance is %f \n", mean(h1(X1)), var(h1(X1))))
cat(sprintf("Integral of h2(x) using normal distribution is %f and variance is %f \n", mean(h2(X2)), var(h2(X2))))

X3 <- 2 - X2
cat(sprintf("Integral of h2(x) using normal distribution is %f and variance is %f \n",
            (mean(h2(X3)) + mean(h2(X2))) / 2,
            (var(h2(X3)) +
              var(h2(X2)) +
              2 * cov(h2(X2), h2(X3))) / 4))

X4 <- -X1
cat(sprintf("Integral of h2(x) using normal distribution is %f and variance is %f",
            (mean(h1(X1)) + mean(h1(X4))) / 2,
            (var(h1(X1)) +
              var(h1(X4)) +
              2 * cov(h1(X1), h1(X4))) / 4))
```

## K-th moment of a uniform random variable on [0, 2]

```{r}
k <- 1:10
moment <- round(2^k / (k + 1), 2)
cat(sprintf("The k-th moment for k ∈ N* of a uniform random variable on [0, 2] is %s", paste(moment, collapse = ", ")))
```


```{r}

```