184 Some Special Distributions3.3.10.Give a reasonable definition of a chi-square distribution with zero degrees
of freedom.
Hint: Work with the mgf of a distribution that isχ^2 (r)andletr=0.3.3.11.Using the computer, obtain plots of the pdfs of chi-squared distributions
with degrees of freedomr=1, 2 , 5 , 10 ,20. Comment on the plots.
3.3.12.Using the computer, plot the cdf of a Γ(5,4) distribution and use it to guess
the median. Confirm it with a computer command that returns the median [In R,
use the commandqgamma(.5,shape=5,scale=4)].3.3.13.Using the computer, obtain plots of beta pdfs forα=1, 5 ,10 andβ=
1 , 2 , 5 , 10 ,20.3.3.14.In a warehouse of parts for a large mill, the average time between requests
for parts is about 10 minutes.(a) Find the probability that in an hour there will be at least 10 requests for
parts.(b) Find the probability that the 10th request in the morning requires at least 2
hours of waiting time.3.3.15.LetXhave a Poisson distribution with parameterm.Ifmis an experi-
mental value of a random variable having a gamma distribution withα=2and
β= 1, computeP(X=0, 1 ,2).
Hint: Find an expression that represents the joint distribution ofXandm.Then
integrate outmto find the marginal distribution ofX.3.3.16.LetXhave the uniform distribution with pdff(x)=1, 0 <x<1, zero
elsewhere. Find the cdf ofY=− 2logX. What is the pdf ofY?3.3.17.Find the uniform distribution of the continuous type on the interval (b, c)
that has the same mean and the same variance as those of a chi-square distribution
with 8 degrees of freedom. That is, findbandc.
3.3.18.Find the mean and variance of theβdistribution.
Hint: From the pdf, we know that
∫ 1
0yα−^1 (1−y)β−^1 dy=Γ(α)Γ(β)
Γ(α+β)for allα> 0 ,β>0.
3.3.19.Determine the constantcin each of the following so that eachf(x)isaβ
pdf:
(a)f(x)=cx(1−x)^3 , 0 <x<1, zero elsewhere.(b)f(x)=cx^4 (1−x)^5 , 0 <x<1, zero elsewhere.(c)f(x)=cx^2 (1−x)^8 , 0 <x<1, zero elsewhere.