3.3. TheΓ,χ^2 ,andβDistributions 1833.3.3.Suppose the lifetime in months of an engine, working under hazardous con-
ditions, has a Γ distribution with a mean of 10 months and a variance of 20 months
squared.(a)Determine the median lifetime of an engine.(b)Suppose such an engine is termed successful if its lifetime exceeds 15 months.
In a sample of 10 engines, determine the probability of at least 3 successful
engines.3.3.4.LetXbe a random variable such thatE(Xm)=(m+1)!2m,m=1, 2 , 3 ,....
Determine the mgf and the distribution ofX.
Hint:Write out the Taylor series^6 of the mgf.3.3.5.Show that
∫∞μ1
Γ(k)zk−^1 e−zdz=k∑− 1x=0μxe−μ
x!,k=1, 2 , 3 ,....This demonstrates the relationship between the cdfs of the gamma and Poisson
distributions.
Hint:Either integrate by partsk−1 times or obtain the “antiderivative” by showing
that
d
dz⎡
⎣−e−zk∑− 1j=0Γ(k)
(k−j−1)!zk−j−^1⎤
⎦=zk−^1 e−z.3.3.6.LetX 1 ,X 2 ,andX 3 be iid random variables, each with pdff(x)=e−x,
0 <x<∞, zero elsewhere.
(a)Find the distribution ofY= minimum(X 1 ,X 2 ,X 3 ).
Hint: P(Y≤y)=1−P(Y>y)=1−P(Xi>y,i=1, 2 ,3).(b)Find the distribution ofY=maximum(X 1 ,X 2 ,X 3 ).3.3.7.LetXhave a gamma distribution with pdf
f(x)=1
β^2
xe−x/β, 0 <x<∞,zero elsewhere. Ifx= 2 is the unique mode of the distribution, find the parameter
βandP(X< 9 .49).
3.3.8. Compute the measures of skewness and kurtosis of a gamma distribution
that has parametersαandβ.
3.3.9. LetXhave a gamma distribution with parametersαandβ. Show that
P(X≥ 2 αβ)≤(2/e)α.
Hint: Use the result of Exercise 1.10.5.
(^6) See, for example, the discussion on Taylor series inMathematical Commentsreferenced in the
Preface.