58 Probability and Distributions1.7.4.Given∫
C[1/π(1 +x(^2) )]dx,whereC⊂C={x:−∞<x<∞}. Show that
the integral could serve as a probability set function of a random variableXwhose
space isC.
1.7.5.Let the probability set function of the random variableXbe
PX(C)=
∫
C
e−xdx, whereC={x:0<x<∞}.
LetCk={x:2− 1 /k < x≤ 3 },k=1, 2 , 3 ,.... Find the limits limk→∞Ckand
PX(limk→∞Ck). FindPX(Ck) and show that limk→∞PX(Ck)=PX(limk→∞Ck).
1.7.6.For each of the following pdfs ofX, findP(|X|<1) andP(X^2 <9).
(a)f(x)=x^2 / 18 ,− 3 <x<3, zero elsewhere.
(b)f(x)=(x+2)/ 18 ,− 2 <x<4, zero elsewhere.
1.7.7.Letf(x)=1/x^2 , 1 <x<∞, zero elsewhere, be the pdf ofX.IfC 1 ={x:
1 <x< 2 }andC 2 ={x:4<x< 5 }, findPX(C 1 ∪C 2 )andPX(C 1 ∩C 2 ).
1.7.8.Amodeof the distribution of a random variableX is a value ofxthat
maximizes the pdf or pmf. If there is only one suchx, it is called themode of the
distribution. Find the mode of each of the following distributions:
(a)p(x)=(^12 )x,x=1, 2 , 3 ,..., zero elsewhere.
(b)f(x)=12x^2 (1−x), 0 <x<1, zero elsewhere.
(c)f(x)=(^12 )x^2 e−x, 0 <x<∞, zero elsewhere.
1.7.9.The median and quantiles, in general, are discussed in Section 1.7.1. Find
the median of each of the following distributions:
(a)p(x)=x!(44!−x)!(^14 )x(^34 )^4 −x,x=0, 1 , 2 , 3 , 4 ,zero elsewhere.
(b)f(x)=3x^2 , 0 <x<1, zero elsewhere.
(c)f(x)=π(1+^1 x (^2) ),−∞<x<∞.
1.7.10.Let 0<p<1. Find the 0.20 quantile (20th percentile) of the distribution
that has pdff(x)=4x^3 , 0 <x<1, zero elsewhere.
1.7.11.For each of the following cdfsF(x), find the pdff(x)[pmfinpart(d)],
the first quartile, and the 0.60 quantile. Also, sketch the graphs off(x)andF(x).
May use R to obtain the graphs. For Part(a) the code is provided.
(a)F(x)=^12 +^1 πtan−^1 (x),−∞<x<∞.
x<-seq(-5,5,.01); y<-.5+atan(x)/pi; y2<-1/(pi*(1+x^2))
par(mfrow=c(1,2));plot(y~x);plot(y2~x)