## Problem 1 
a <- 2
reps <- 10000

## Generate a single N(0,1) random draw, 10000 times
Z <- rnorm(reps)

## Find the variance of these draws
var(Z)

## Find the variance of a times each draw
var(a*Z)

# Verify that the var(a*Z) = a^2 * var(Z)
a^2*var(Z)

## Problem 2

n <- 4
reps <- 10000

## Generate 4 iid N(0,1) random draws 10000 times
Z <- array( rnorm(n*reps), c(n, reps) )

## Find the variances of the 4 iid random draws
fourVariances <- apply(Z, 1, var)

## Verify that the variance of the sum is roughly the sum of the variances
varOfSum <- var( apply(Z, 2, sum) )

print(varOfSum)
print(sum(fourVariances))

## Problem 5

V <- NULL
reps <- 10000
for (lambda in c(1,2,3,4))
{
  for (n in c(10,20,40,80))
  {
    X <- array(rexp(n*reps, lambda), c(n,reps))
    M <- apply(X, 2, mean)
    V <- rbind(V, c(lambda, n, var(M), 1/(n*lambda^2)))
  }
}


## Problem 8

V <- NULL
reps <- 10000
for (lambda in c(1,2,3,4))
{
  for (n in c(10,20,40,80))
  {
    X <- array(rpois(n*reps, lambda), c(n,reps))
    M <- apply(X, 2, mean)
    V <- rbind(V, c(lambda, n, var(M), lambda/n))
  }
}