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Chapter 5: Q. 5.20 (page 218)

For any real number y, define y+by

y+=y,ify鈮�00,ify<0

Let cbe a constant.

(a) Show that

E(Z-c)+=12蟺e-c2/2-c(1-桅(c))

when Zis a standard normal random variable.

(b) Find E(X-c)+when Xis normal with mean 渭and variance 蟽2.

Short Answer

Expert verified

a. Using the concept of a random variable's expectation it is proved.

b. The value of E(X-c)+is1蟽脳12蟺e-c-渭蟽22-c-渭蟽1-桅c-渭蟽.

Step by step solution

01

Random variable

Observe that the random variable (Z-c)+is, in fact, a random variable.

(Z-c)+(蝇)=Z(蝇)-c,Z(蝇)鈮�c0,Z(蝇)<c

for蝇鈭�蝇, where probability space is(蝇,F,P).

02

Showing of standard normal random variable (part a)

a.

Intended consequence is,

E(Z-c)+=鈭�惟(Z-c)+(蝇)dP(蝇)

=鈭�mtight" style="margin-right:0.07153em;">Z(蝇)鈮�c(Z(蝇)-c)dP(蝇)

Density function,

蠁(z)=12蟺e-z22

So integral is,

鈭�Z(蝇)鈮�c(Z(蝇)-c)dP(蝇)=鈭�z鈮�c(z-c)蠁(z)dz

=鈭�c鈭�锄蠁(z)dz-c鈭�c鈭�蠁(z)dz

First integral,

u=z22鈬�du=zdz

Substitute values we get,

鈭�c鈭�锄蠁(z)dz=12蟺鈭�c鈭�ze-z22dz

=12蟺鈭�c22鈭�e-udu

=12蟺e-c22

Second integral,

E(Z-c)+=12蟺e-c22-c(1-桅(c))

03

Calculation of E(X-c)+ (part b)

b.

For,

X~N渭,蟽2.

Hence Z=X-渭蟽~N(0,1).

So, X=蟽窜+渭.

(X-c)+=(蟽窜+渭-c)+

=1蟽Z-c-渭蟽+

since 蟽>0.

Using part (a) we get,

E(X-c)+=1蟽EZ-c-渭蟽+

=1蟽脳12蟺e-c-渭蟽22-c-渭蟽1-桅c-渭蟽

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