```{r setup, include=FALSE}
knitr::opts_chunk$set(
  echo = TRUE,comment = '' 
  )
```


--- 

## Prerequisites  
### Installing and loading packages

```
install.packages("ggplot2")
```

```{r}
library(ggplot2) # graphics library
```

## Problem 1: $N(0,1)$

* Let  us pick sample size $n=10$. Let $X\sim N(0,1)$. 
* Repeat  the following  $10,000$  times: 
  * Draw a random sample $X_1,\dots,X_n$ from the  normal  density $f(x)$.  
  * For $k\in\{1,3,5,9,10\}$, compute the $k$-th statistic $y_k$.  
* Plot the histogram of  $y_k$ for each $k$. 

```{r}
f <- function(n.sim, n, k=1:n, p=pnorm, d=dnorm, r=rnorm, name, ...) {
  if (missing(name)) name <- ""
  k <- sort(unique(k))[1 <= k & k <= n]

  # Perform the simulation.
  sim <- apply(matrix(r(n.sim*n, ...), nrow=n), 2, sort)[k, , drop=FALSE]

  # Plot the requested order statistics.
  for (i in 1:length(k)) {

    # Plot the empirical distribution.
    hist(sim[i, ], freq=FALSE, 
         xlab="Value",
         sub=paste(n.sim, "iterations with sample size", n),
         main=paste(name, "order statistic", k[i]))
  }
}
#
# Set up to simulate and display the results.
#
par(mfrow=c(2,4))
n.sim <- 1e4
set.seed(17)

# Study Normal order statistics.
f(n.sim, 10, c(1,3,5,9), name="Normal(0,1)")
```




## Problem 2: $Gamma(1.5,5)$

* Let  us pick sample size $n=25$. Let $X\sim Gamma(1.5,5)$. 
* Repeat  the following  $10,000$  times: 
  * Draw a random sample $X_1,\dots,X_n$ from the  normal  density $f(x)$.  
  * For $k\in\{2,4,16,24\}$ compute the $k$-th statistic $y_k$.  
* Plot the histogram of  $y_k$ for each $k$. 

```{r}
#
# Set up to simulate and display the results.
#
par(mfrow=c(2,4))
n.sim <- 1e4
set.seed(17)

# Study Gamma order statistics.
# This illustrates how to use `f` for general distributions.
f(n.sim, 25, c(2, 4, 16, 24), name="Gamma(1.5,5)", 
  p=pgamma, d=dgamma, r=rgamma, shape=1.5, scale=5)
```




# Acknowledgement

Sourced from
[whuber](https://stats.stackexchange.com/users/919/whuber), Simulating order statistics, URL (version: 2017-04-13): [see post](https://stats.stackexchange.com/q/241816)

    


