Problem 12.2.Rmd

---
title: "12.2 Independent sampling"
output: html_notebook
---
#12.2.1
```{r}
fPDF <- function(X){
  y <- ifelse(X < 0.9735, (1 / 1.33485) * (1 / sqrt(2 * pi)) * exp(-X ^ 2 / 2), ifelse(X <= 5, 0.186056, 0))
return(y)
}

integrate(fPDF,0,8)

```

```{r}
fReject <- function(N){
  count <- 1
  lSamples <- vector(length=N)
  while(count <= N){
    X <- runif(1, 0, 5)
    Y <- runif(1, 0, (1 / 1.335) * (1 / sqrt(2 * pi)))
    if(Y < fPDF(X)){
      lSamples[count] <- X
      count <- count + 1
    }
}
  return(lSamples)
}
mean(fReject(10000))
var(fReject(10000))
```
```{r}
fIntegrator <- function(X){
  return(integrate(fPDF, 0, X)[[1]])
}
lCDF <- sapply(seq(0, 5, 0.1), fIntegrator)
fICDF <- approxfun(lCDF, seq(0, 5, 0.1))
fInverseTransform <- function(N){
  lCDF <- runif(N, 0, 1)
  return(sapply(lCDF, fICDF))
}
mean(fInverseTransform(100000))
var(fInverseTransform(100000))
```

```{r}
fImportance <- function(N){
  lX <- runif(N, 0, 5)
  lF <- sapply(lX, fPDF)
  lG <- rep(1 / 5, N)
  lRatio <- lX * lF / lG
  mean(lRatio)
}
fImportance(10000)
```
```{r}
rep(1 / 5, 20)
```