/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q.25 Show that if X聽and Y聽are indep... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

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

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider a gambler who, at each gamble, either wins or loses her bet with respective probabilities pand 1-p. A popular gambling system known as the Kelley strategy is to always bet the fraction 2p-1of your current fortune when p>12. Compute the expected fortune afterngambles of a gambler who starts with xunits and employs the Kelley strategy.

There are n items in a box labeled H and m in a box labeled T. A coin that comes up heads with probability p and tails with probability 1 鈭� p is flipped. Each time it comes up heads, an item is removed from the H box, and each time it comes up tails, an item is removed from the T box. (If a box is empty and its outcome occurs, then no items are removed.) Find the expected number of coin flips needed for both boxes to become empty. Hint: Condition on the number of heads in the first n + m flips.

A die is rolled twice. Let X equal the sum of the outcomes, and let Y equal the first outcome minus the second.

ComputeCov(X,Y).

If 10 married couples are randomly seated at a round table, compute

(a) The expected number and

(b) The variance of the number of wives who are seated next to their husbands.

Let Xbe the number of 1鈥�sand Ythe number of 2'sthat occur in nrolls of a fair die. Compute Cov(X,Y).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.