Question

1) Drawt (b) The normal f. with -50, ơ-10 (d) The expogEntial Ad.f with parameter λ raphs ofthe p.d.f. of the following distributions S (a) The standard no mal p.d.f. Ing (c) The unifo f. over interval [10, 20] 2. 2) Illustrating the central limit theorem Let X be a random variable having the exponential distribution with A-2. Denote by X,, X,, Xj, a sequence of independent variables with the same distribution as X. Define the sample mean x by The central limit theorem applied to this particular case implies that the probability distribution of o/Vn converges to the standard normal distribution for certain values of and ơ (a) For what values of μ and σ does the convergence hold? (b) For each of the four values n = 5, 10, 20 and 50 do the following: Obtain 10,000 independent values of the sample mean random variable X where the X, are generated from the exponential distribution with λ 2; then create a histogram of olNn for each n. Discuss what you notice for each case on n. 3) Testing the normal approximation with Q-Q plots. We e seen class lectures how normal plots (use qqnorm command in R) (which are a special case of Q-Q plots) provide another way to decide whether given data are roughly normally distributed. For the same ơAn as in the above problem, and for each n -5, 10, 20 and 50, obtain the normal probability plots (using qqnorm command). Does it look like the sample means are becoming more normally distributed as n increases? Discuss what you observe Using R #2 and #3 need to be answered

Answer 1

2a) We assume that the distribution has rate $\lambda=2$. Then

$E(X)=1/\lambda,~~Var(X)=1/\lambda^2$

Thus from CLT we get

$\mu=E(X)=1/\lambda,~~\sigma=\sqrt{Var(X)}=1/\lambda$

2b) R Program

##Question 2(b)

la=2
rep=10000
w=numeric(rep)
f=function(n)
{
for(i in 1:rep)
{
x=rexp(n,rate=2)
w[i]=sqrt(n)*(mean(x)-(1/2))/(1/2)
}
hist(w,main="Histogram",col="red",xlab="")
text(4,1225,labels=n)
}
par(mfrow=c(2,2))
f(5)
f(10)
f(20)
f(50)

R OUTPUT

Histogram Histogram 10 -20246 -20246 Histogram Histogram 20 50 2 0 2 4 2 02 4

From the histogram (the numerical figure on the graph indicates value of n), we find that as n increases, histogram becomes closer to that of a standard normal distribution.

3. R Program

## Question 3

la=2
rep=10000
w=numeric(rep)
g=function(n)
{
for(i in 1:rep)
{
x=rexp(n,rate=2)
w[i]=sqrt(n)*(mean(x)-(1/2))/(1/2)
}
qqnorm(w,main="Q-Q Plot",col="green",xlab="")
text(-3,3,labels=n)
}
par(mfrow=c(2,2))
g(5)
g(10)
g(20)
g(50)

R OUTPUT

Q-Q Plot Q-Q Plot 寸 10 4 2024 4 2 02 4 Q-Q Plot Q-Q Plot 寸 20 50 4 2 024 4 2 02 4

From the Q-Q plot(the numerical figure on the graph indicates value of n), we find that as n increases, more and more points are concentrating on a straightline. Hence the distribution becomes close to normal with increase in n.