1.  generic

2.  

varDivByN <- function(x)    # x is vector of sample
{
   vr <- var(x)
   n <- length(x)
   ((n-1)/n) * vr
}

print(varDivByN(c(5,12,13,8.88)))  # 9.74

3.

varHist <- function(n,i)
{
   # N_i has a binomial distribution, so Var(N_i) = np(-1p);
   # here n = sample size and # p = P(i-1 < X < i); 
   p <- pexp(i,1/2) - pexp(i-1,1/2)
   n*p*(1-p)
}

print(varHist(100,2))  # 18.17

4.


mmGamma <- function(x)  {

   xbar <- mean(x)
   s2 <- var(x)
   estR <- xbar^2 / s2
   estLambda <- xbar / s2
   return(c(estR,estLambda))
}

print(mmGamma(c(5,12,13,8.88)))  # 7.28, 7.49