Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Q.7.30 (page 361)

Consider Example 4f, which is concerned with the multinomial distribution. Use conditional expectation to compute E[NiNj], and then use this to verify the formula for Cov(Ni, Nj) given in Example 4f.

Short Answer

Expert verified

Therefore it has been computed that ENiNj=m(m−1)pipjby using conditional expectation

Step by step solution

01

Given information

Use conditional expectation to compute E[NiNj],

02

Solution

Compute the ENiNjusing conditional expectation.

So,

ENiNj∣Ni=NiENj∣Ni

=Nim−Nipj1−pi

Now, each of the m-Ni trails no resulting in outcome i will independently result inj with probability pj1-pi.

03

Solution

Therefore,

ENiNj=pj1−pimENi−ENi2

=pj1−pim2pi−m2pi2−mpi1−pi

=pj1−pim(m−1)pi1−pi

=m(m−1)pipj

And,

Cov⁡Ni,Nj=m(m−1)pipj−m2pipj

=m2pipj−mpipj−m2pipj

=−mpipj

04

Final answer 

Therefore it has been computed that ENiNj=m(m−1)pipjby using conditional expectation

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

The county hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital, whose coordinates are (0,0), to the point(x,y) is |x|+|y|. If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance.

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.

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 − 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 fair die is successively rolled. Let X and Y denote, respectively, the number of rolls necessary to obtain a 6 and a 5. Find

(a) E[X];

(b) E[X∣Y=1];

(c) E[X∣Y=5];

Suppose that the expected number of accidents per week at an industrial plant is 5. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2.5. If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week .
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.

Sign-up for free