Question

Simulate n values of an exponential random variable X with parameter λ (of your choice), and compute the sample mean i, sample median m, sample standard deviation s. Plot these quantities as functions of n (on three separate plots). Do x, m, and s converge to any limit values, as n-oo? What are those values and how are they related? Estimate the variance of both x and m for a particular value of n, such as n 100 (by generating, say, 1000 random samples of size n and computing the sample variance of the resulting estimates iand m).

Answer 1

0 200 600 800 1000

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O o O o ふ。。 o O 0 200 400 600 800 1000

Here we take, X 2 rA exp (-rA) dr 0 : Var(X) = E(X2)-E2(X) SD(X) CDF of X is Fr(z) = P(X < x) = 1-e- log 2 Here we take, λ 2 then T, s converge to 0.5 whereas m converges to 0.35. These can be shown in the above graphs and also from the simulation outputs (see below the R codes and outputs).

#Lambda=2(population mean=1/2)
n=100
t=1000
x=matrix(0,nrow=t,ncol=n)
m1=1:t*0 # sample mean
m=1:t*0 # sample median
s=1:t*0 # sample sd
for(i in 1:t)
{
x[i,]=rexp(n,rate=2)
m1[i]=mean(x[i,])
m[i]=median(x[i,])
s[i]=sd(x[i,])
}
plot(m1,lwd=2,type="p",xlab="",ylab="sample means")
abline(h=0.5,lwd=2,col=2)
plot(m,lwd=2,type="p",xlab="",ylab="sample medians")
abline(h=0.35,lwd=2,col=2)
plot(s,lwd=2,type="p",xlab="",ylab="sample sds")
abline(h=0.5,lwd=2,col=2)
round(mean(m1),4)
round(mean(m),4)
round(mean(s),4)

Outputs:

> round(mean(m1),4)
[1] 0.5005
> round(mean(m),4)
[1] 0.3506
> round(mean(s),4)
[1] 0.4943