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Chapter 7: Q.26 (page 361)

Prove thatE[g(X)Y∣X]=g(X)E[Y∣X].

Short Answer

Expert verified

We prove that,E[g(X)Y∣X]=g(X)E[Y∣X]

Step by step solution

01

Given information

Given in the question that, we have to prove thatE[g(X)Y∣X]=g(X)E[Y∣X]

02

Explanation

Because of the clarity, suppose that Xand Yare continuous random variables with its density functions. Take any x∈suppfX. We have that

E(g(X)Y∣X=x)=∬ℝ2 g(x)yf(X,Y)∣X=x(x,y)dxdy

But, we have that

f(X,Y)∣X=x(x,y)dxdy=fY∣X(y∣x)dy

So the integral becomes

∫ℝ g(x)yfY∣X(y∣x)dy

Which is equal to

g(X)E(Y∣X=x)

Since the relation hold for every x∈suppf, we end up with

E(g(X)Y∣X)=g(X)E(Y∣X)

03

Final answer

We prove that,E[g(X)Y∣X]=g(X)E[Y∣X]

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Most popular questions from this chapter

Consider a graph having nvertices labeled1,2,...,n, and suppose that, between each of the n2pairs of distinct vertices, an edge is independently present with probability p. The degree of a vertexi, designated asDi,is the number of edges that have vertex ias one of their vertices.

(a) What is the distribution of Di?

(b) Find ρ(Di,Dj), the correlation between DiandDj.

Suppose that A and B each randomly and independently choose3of10objects. Find the expected number of objects

a. Chosen by both A and B;

b. Not chosen by either A or B;

c. Chosen by exactly one of A and B.

A group of nmen and n women is lined up at random.

(a) Find the expected number of men who have a woman next to them.

(b) Repeat part (a), but now assuming that the group is randomly seated at a round table.

The joint density of X and Y is given by

f(x,y)=12πe-ye-(x-y)2/20<y<∞,

-∞<x<∞

(a) Compute the joint moment generating function of X and Y.

(b) Compute the individual moment generating functions.

Consider the following dice game, as played at a certain gambling casino: Players1and 2roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Playeri,i=1,2,wins if his roll is strictly greater than the banks. Fori=1,2,let

Ii=1    ifiwins0    otherwise

and show that I1and I2are positively correlated. Explain why this result was to be expected.
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