Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

60 Probability and Distributions

1.7.20.The distribution of the random variableXin Example 1.7.3 is often used

to model the log of the lifetime of a mechanical or electrical part. What about the

lifetime itself? LetY=exp{X}.

(a)Determine the range ofY.

(b)Use the transformation technique to find the pdf ofY.

(c)Write an R function to compute this pdf and use it to obtain a graph of the

pdf. Discuss the plot.

(d)Determine the 90th percentile ofY.

1.7.21.The distribution of the random variableXin Example 1.7.3 is a member

of the log-Ffamilily. Another member has the cdf

F(x)=

[

1+

2

3

e−x

]− 5 / 2

, −∞<x<∞.

(a)Determine the corresponding pdf.

(b)Write an R function that computes this cdf. Plot the function and obtain

approximations of the quartiles and median by inspection of the plot.

(c)Obtain the inverse of the cdf and confirm the percentiles in Part(b).

1.7.22.LetXhave the pdff(x)=x^2 / 9 , 0 <x<3, zero elsewhere. Find the pdf

ofY=X^3.

1.7.23.If the pdf ofXisf(x)=2xe−x

2
, 0 <x<∞, zero elsewhere, determine
the pdf ofY=X^2.

1.7.24.LetXhave the uniform pdffX(x)=^1 π,for−π 2 <x<π 2 .Findthepdfof

Y=tanX. This is the pdf of aCauchy distribution.

1.7.25.LetXhave the pdff(x)=4x^3 , 0 <x<1, zero elsewhere. Find the cdf
and the pdf ofY=−lnX^4.

1.7.26.Letf(x)=^13 ,− 1 <x<2, zero elsewhere, be the pdf ofX.Findthecdf
and the pdf ofY=X^2.
Hint:ConsiderP(X^2 ≤y) for two cases: 0≤y<1and1≤y<4.

1.8 ExpectationofaRandomVariable

In this section we introduce the expectation operator, which we use throughout

the remainder of the text. For the definition, recall from calculus that absolute

convergence of sums or integrals implies their convergence.