Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.4. The Normal Distribution 197

(a)Clearlyf(x;α)>0foallx. Show that the pdf integrates to 1 over (−∞,∞).

Hint:Start with

∫∞

−∞

f(x;α)dx=2

∫∞

−∞

φ(x)

∫αx

−∞

φ(t)dt.

Next sketch the region of integration and then combine the integrands and

use the polar coordinate transformation we used after expression (3.4.1).

(b)Note thatf(x;α)istheN(0,1) pdf forα= 0. The pdfs are left skewed for

α<0 and right skewed forα>0. Using R, verify this by plotting the pdfs

forα=− 3 ,− 2 ,− 1 , 1 , 2 ,3. Here’s the code forα=−3:

x=seq(-5,5,.01); alp =-3; y=2*dnorm(x)*pnorm(alp*x);plot(y~x)

This family is called theskewed normal family; see Azzalini (1985).

3.4.28.ForZdistributedN(0,1), it can be shown that

E[Φ(hZ+k)] = Φ[k/

√

1+h^2 ];

see Azzalini (1985). Use this fact to obtain the mgf of the pdf (3.4.20). Next obtain
the mean of this pdf.

3.4.29. LetX 1 andX 2 be independent with normal distributionsN(6,1) and
N(7,1), respectively. FindP(X 1 >X 2 ).
Hint: WriteP(X 1 >X 2 )=P(X 1 −X 2 >0) and determine the distribution of
X 1 −X 2.

3.4.30.ComputeP(X 1 +2X 2 − 2 X 3 >7) ifX 1 ,X 2 ,X 3 are iid with common

distributionN(1,4).

3.4.31.A certain job is completed in three steps in series. The means and standard
deviations for the steps are (in minutes)

Step Mean Standard Deviation

117 2

213 1

313 2

Assuming independent steps and normal distributions, compute the probability that

the job takes less than 40 minutes to complete.

3.4.32.LetXbeN(0,1). Use the moment generating function technique to show

thatY=X^2 isχ^2 (1).

Hint: Evaluate the integral that representsE(etX

2

)bywritingw=x

√

1 − 2 t,

t<^12.

3.4.33.SupposeX 1 ,X 2 are iid with a common standard normal distribution. Find
the joint pdf ofY 1 =X 12 +X 22 andY 2 =X 2 and the marginal pdf ofY 1.
Hint: Note that the space ofY 1 andY 2 is given by−
√
y 1 <y 2 <
√
y 1 , 0 <y 1 <∞.