Question

Question 2

Please submit worked solutions, including a copy of Stata commands and graphs to the following questions.

(a) Suppose a random variable X has pdf f(x) = 2xe−x2, where x > 0. Draw a random sample of size 10000 from this distribution. Draw a histogram of the 10000 sampled values and obtain the sample mean and standard deviation.
(b) Using your sample obtained above, draw a histogram and calculate mean and variance for 10000 sampled values for the random variable Y = X2.
(c) From the pdf f(x) = 2xe−x2, where x > 0, find the distribution of Y = X2. The distribution of Y is a special case of which distribution? Hence, what are E(Y ) and Var(Y)? [Hint: look inside back cover of WMS!] Compare these values with your sample results above.

Answer 1

a)

f(x) = 2xe−x2, where x > 0 it is the pdf of Weibull distribution with parameter ( =1, k=2) the pdf of weibull distribution is

command of r software

sample=rweibull(10000,2,1)
histo=hist(sample)
histo

> mean=mean(sample)
> std=sd(sample)
> mean
[1] 0.8845851
> std
[1] 0.4665082

b)

> y=sample^2
> mean(y)
[1] 1.000099
> sd(y)
[1] 1.011358

> histo2=hist(y)

c)

E(Y)=1

Var(Y)=1