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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 a group of 100 people, compute

(a) the expected number of days of the year that are birthdays of exactly 3 people;

(b) the expected number of distinct birthdays.

Show that E[(X鈭�a)2] is minimized at a=E[X].

Let Z be a standard normal random variable,and, for a 铿亁ed x, set

X={ZifZ>x0otherwise

Show thatE[X]=12蟺e鈭�x2/2.

Let 蠒be the standard normal distribution function, and let X be a normal random variable with mean 渭 and variance 1. We want to find E[ 蠒(X)]. To do so, let Z be a standard normal random variable that is independent of X, and let

I=1,鈥呪赌呪赌呪赌�ifZ<X0,鈥呪赌呪赌呪赌�ifZ鈮�X

(a) Show that E[I鈭�X=x]=桅(x).

(b) Show that E[桅(X)]=P{Z<X}.

(c) Show that E[桅(X)]=桅渭2.

Hint: What is the distribution of X-Z?

The preceding comes up in statistics. Suppose you are about to observe the value of a random variable X that is normally distributed with an unknown mean 渭 and variance 1, and suppose that you want to test the hypothesis that the mean 渭 is greater than or equal to 0. Clearly you would want to reject this hypothesis if X is sufficiently small. If it results that X = x, then the p-value of the hypothesis that the mean is greater than or equal to 0 is defined to be the probability that X would be as small as x if 渭 were equal to 0 (its smallest possible value if the hypothesis were true). (A small p-value is taken as an indication that the hypothesis is probably false.) Because X has a standard normal distribution when 渭 = 0, the p-value that results when X = x is 蠒 (x). Therefore, the preceding shows that the expected p-value that results when the true mean is 渭 is 蠁渭2 .

Let X1,X2,鈥�,Xnbe independent random variables having an unknown continuous distribution function Fand let Y1,Y2,鈥�,Ymbe independent random variables having an unknown continuous distribution function G. Now order those n+mvariables, and let

Ii=1鈥呪赌呪赌呪赌�if theith smallest of then+m鈥呪赌呪赌呪赌�variables is from theXsample0鈥呪赌呪赌呪赌�otherwise

The random variable R=鈭�i=1n+miIiis the sum of the ranks of the Xsample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether Fand Gare identical distributions. This test accepts the hypothesis that F=Gwhen Ris neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R.

Hint: Use the results of Example 3e.
See all solutions

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