Question

Exercise 5 (More Simulation) Let X follow an exponential distribution with rate parameter λx = 2. Let Y follow a Poisson distribution with rate parameter λ 3. We write sd(X) for the true standard deviation of X and m(Y) for the true median of Y Let s be the sample standard median which is an estimate of m(Y) Suppose we take samples of size nz -10 from X and take samples of size ny 5. Consider the statistic of X which is an estimate of sd(X). Also let my be the sample 8 my what is the (sampling) distribution of ? We could ask a statistician who specializes in theory. Instead of using mathematics, simulate 5000 times and store the results. Plot a histogram of the observed values of SL Comment on the shape of the histogram and ennpirical distribution of s Before running your code, set mly my my the same seed used for the previous exercise. For full credit, do not use a for loop.

Please complete using R. Show all code needed to complete exercise. Will Thumbs up if done neatly and correctly.

Answer 1

REQUIRED R CODE

set.seed(100)
n<-5000
a<-0
ans<-c()
while(a<n)
{
sx<-sd(rexp(n = 10,rate = 2))
my<-median(rpois(n = 5,lambda = 3))
ans<-c(ans,sx/my)
a<-a+1
}
hist(x = ans,xlim = c(0,0.6),breaks = 200)

empirically standard deviation of x follows normal

also median of poisson follows normal so

this ration is ration of two normal distributions

and the histogram observed is negatively skewed

here the standard deviation of exp(2) is 0.5 and median of poisson(3) is 3 so this distribution in expected value is 0.5/3 i.e 1.666.