Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

196 Some Special Distributions

3.4.21.Letf(x)andF(x) be the pdf and the cdf, respectively, of a distribution of
the continuous type such thatf′(x) exists for allx. Let the mean of the truncated
distribution that has pdf g(y)=f(y)/F(b), −∞<y<b, zero elsewhere, be
equal to−f(b)/F(b) for all realb.Provethatf(x) is a pdf of a standard normal
distribution.

3.4.22.LetXandYbe independent random variables, each with a distribution
that isN(0,1). LetZ=X+Y. Find the integral that represents the cdfG(z)=
P(X+Y≤z)ofZ. Determine the pdf ofZ.
Hint: We have thatG(z)=

∫∞

−∞H(x, z)dx,where

H(x, z)=

∫z−x

−∞

1

2 π

exp[−(x^2 +y^2 )/2]dy.

FindG′(z) by evaluating

∫∞
−∞[∂H(x, z)/∂z]dx.
3.4.23.SupposeXis a random variable with the pdff(x) which is symmetric
about 0; i.e.,f(−x)=f(x). Show thatF(−x)=1−F(x), for allxin the support
ofX.

3.4.24.Derive the mean and variance of a contaminated normal random variable.
They are given in expression (3.4.16).

3.4.25.Investigate the probabilities of an “outlier” for a contaminated normal ran-
dom variable and a normal random variable. Specifically, determine the probability
of observing the event{|X|≥ 2 }for the following random variables (use the R
functionpcnfor the contaminated normals):

(a)Xhas a standard normal distribution.

(b)Xhas a contaminated normal distribution with cdf (3.4.15), where =0. 15

andσc= 10.

(c)Xhas a contaminated normal distribution with cdf (3.4.15), where =0. 15

andσc= 20.

(d)Xhas a contaminated normal distribution with cdf (3.4.15), where =0. 25

andσc= 20.

3.4.26.Plot the pdfs of the random variables defined in parts (a)–(d) of the last

exercise. Obtain an overlay plot of all four pdfs also. In R the domain values of the

pdfs can easily be obtained by using theseqcommand. For instance, the command

x<-seq(-6,6,.1)returns a vector of values between−6 and 6 in jumps of 0.1.

Then use the R functiondcnfor the contaminated normal pdfs.

3.4.27.Consider the family of pdfs indexed by the parameterα,−∞<α<∞,

given by

f(x;α)=2φ(x)Φ(αx), −∞<x<∞, (3.4.20)

whereφ(x)andΦ(x) are respectively the pdf and cdf of a standard normal distri-
bution.