Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

194 Some Special Distributions

see Exercise 3.4.24. Upon differentiating (3.4.15), the pdf ofWis

fW(w)=φ(w)(1− )+φ(w/σc)

σc

, (3.4.17)

whereφis the pdf of a standard normal.
Suppose, in general, that the random variable of interest isX=a+bW,where
b>0. Based on (3.4.16), the mean and variance ofXare

E(X)=aand Var(X)=b^2 (1 + (σc^2 −1)). (3.4.18)

From expression (3.4.15), the cdf ofXis

FX(x)=Φ

(

x−a

b

)

(1− )+Φ

(

x−a

bσc

)

, (3.4.19)

which is a mixture of normal cdfs.
Based on expression (3.4.19) it is easy to obtain probabilities for contami-
nated normal distributions using R. For example, suppose, as above,W has cdf
(3.4.15). ThenP(W ≤w) is obtained by the R command(1-eps)pnorm(w) +
epspnorm(w/sigc),whereepsandsigcdenote andσc, respectively. Similarly,
the pdf ofWatwis returned by(1-eps)dnorm(w) + epsdnorm(w/sigc)/sigc.
The functionspcnanddcn^7 compute the cdf and pdf of the contaminated normal,
respectively. In Section 3.7, we explore mixture distributions in general.

EXERCISES

3.4.1.If

Φ(z)=

∫z

−∞

1

√

2 π

e−w

(^2) / 2
dw,
show that Φ(−z)=1−Φ(z).
3.4.2.IfXisN(75,100), findP(X<60) andP(70<X<100) by using either
Table II or the R commandpnorm.
3.4.3. IfXisN(μ, σ^2 ), findbso thatP[−b<(X−μ)/σ < b]=0.90, by using
either Table II of Appendix D or the R commandqnorm.
3.4.4.LetXbeN(μ, σ^2 )sothatP(X<89) = 0.90 andP(X<94) = 0.95. Find
μandσ^2.
3.4.5.Show that the constantccan be selected so thatf(x)=c 2 −x
2
,−∞<x<
∞, satisfies the conditions of a normal pdf.
Hint: Write 2 =elog 2.
3.4.6.IfXisN(μ, σ^2 ), show thatE(|X−μ|)=σ
√
2 /π.
3.4.7.Show that the graph of a pdfN(μ, σ^2 ) has points of inflection atx=μ−σ
andx=μ+σ.
(^7) Downloadable at the site listed in the Preface.