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Chapter 7: Q.25 (page 361)

Show that if Xand Yare independent, then

E[X鈭�Y=y]=E[X]for ally

(a) in the discrete case;

(b) in the continuous case.

Short Answer

Expert verified

The calculation for both cases is similar. Just use the fact that X and Y are independent.

Step by step solution

01

Given information(part a)

Given in the question that,E[X鈭�Y=y]=E[X]for ally

02

Explanation (part a)

Let's begin from the left side. We have that

E(X鈭�Y=y)=鈭�x鈥�xP(X=x鈭�Y=y)

=鈭�x鈥�xP(X=x)

=E(X)

where the second equality is the consequence of the fact that X and Y are independent.

03

Final answer(part a)

We proved that Xand Yare independent

04

Given information(part b)

Given in the question that,E[X鈭�Y=y]=E[X]for ally

05

Explanation(part b)

Let's again start from the left side. For y鈭�suppfY, we have that

E(X鈭�Y=y)=鈭�鈩�鈥�xfX鈭�Y(x鈭�y)dx

Because of the independence, we have that

fX鈭�Y(x鈭�y)=f(x,y)fY(y)=fX(x)fY(y)fY(y)=fX(x)

so we have that the integral is equal to

鈭�鈩�鈥�xfX鈭�Y(x鈭�y)dx=鈭�鈩�鈥�xfX(x)dx=E(X)

so we have demonstrated the asserted in the two cases.

06

Final answer(part b)

We proved that X and Y are independent

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Most popular questions from this chapter

For an event A, let IA equal 1 if A occurs and let it equal 0 if A does not occur. For a random variable X, show that E[X|A] = E[XIA] P(A

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