# Exercise 8 - a : Truncated Normal Distribution

```{r}
f <- function(x, b, mean, sd) {
  1 / (sqrt(2 * pi * sd^2) * pnorm((mean - b) / sd)) *
    exp(-(x - mean)^2 / (2 * sd^2)) *
    (x >= b)
}

M <- function(b, mean, sd) {
  1 / pnorm((mean - b) / sd)
}

g <- function(x, b, mean, sd) {
  1 / sqrt(2 * pi * sd^2) *
    exp(-(x - mean)^2 / (2 * sd^2))
}

n <- 10000
mean <- 0
sd <- 2
b <- 2

x <- numeric(0)
while (length(x) < n) {
  U <- runif(1)
  X <- rnorm(1, mean, sd)
  x <- append(x, X[U <= (f(X, b, mean, sd) / (M(b, mean, sd) * g(X, b, mean, sd)))])
}

t <- seq(b, 7, 0.01)
hist(x, freq = FALSE, breaks = 35)
lines(t, f(t, b, mean, sd), col = "red", lwd = 2)
```

# Exercise 8 - b : Truncated Exponential Distribution
```{r}
f <- function(x, b, lambda) {
  lambda * exp(-lambda * (x - b)) * (x >= b)
}

M <- function(b, lambda) {
  exp(lambda * b)
}

g <- function(x, lambda) {
  lambda * exp(-lambda * x)
}

n <- 10000
b <- 2
lambda <- 1

x <- numeric(0)
while (length(x) < n) {
  U <- runif(1)
  X <- rexp(1, lambda)
  x <- append(x, X[U <= (f(X, b, lambda) / (M(b, lambda) * g(X, lambda)))])
}

t <- seq(b, 7, 0.01)
hist(x, freq = FALSE, breaks = 35)
lines(t, f(t, b, lambda), col = "red", lwd = 2)
```