/*! 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. 7.5 Let A1,A2,鈥�,An聽be arbitrary e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q. 7.5 (page 359)

Let A1,A2,鈥�,Anbe arbitrary events, and define

Ck={at least kof the Aioccur}. Show that

鈭�k=1nPCk=鈭�k=1nPAk

Hint: Let Xdenote the number of the Aithat occur. Show

that both sides of the preceding equation are equal to E[X].

Short Answer

Expert verified

The arbitrary events is showed as鈭�k=1nPCk=E(X)=鈭�k=1nPAk.

Step by step solution

01

Given Information

PCk=P(X鈮�k)as arbitrary events inA1,A2,鈥�,An.

02

Explanation

We have that,PCk=P(X鈮�k)

Hence,鈭�k=1nPCk=鈭�k=1nP(X鈮�k)=鈭�k=0鈭�P(X>k)=E(X)

where the last equality is the famous expression of the mean of non-negative discrete random variable. On the other hand, define random variables Iito be indicators whether event Aihas occurred or not.

03

Explanation

We have that, X=鈭�k=1nIk

Because of the linearity of the mean, we have that,

E(X)=鈭�k=1nEIk=鈭�k=1nPIk=1=鈭�k=1nPAk

so we have showed that,鈭�k=1nPCk=E(X)=鈭�k=1nPAk.

04

Final answer

The arbitrary events is showed as鈭�k=1nPCk=E(X)=鈭�k=1nPAk.

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

Suppose that X1and X2are independent random variables having a common mean 渭. Suppose also that VarX1=蟽12and VarX2=蟽22. The value of 渭is unknown, and it is proposed that 渭be estimated by a weighted average of X1and X2. That is, role="math" localid="1647423606105" 位X1+(1-位)X2will be used as an estimate of 渭for some appropriate value of 位. Which value of 位yields the estimate having the lowest possible variance? Explain why it is desirable to use this value of 位

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 group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of a man and a woman. Now suppose the 20 people consist of 10 married couples. Compute the mean and variance of the number of married couples that are paired together.

In Problem 7.6, calculate the variance of the sum of the rolls.

In the text, we noted that

E鈭�i=1鈭�Xi=鈭�i=1鈭�EXi

when the Xiare all nonnegative random variables. Since

an integral is a limit of sums, one might expect that E鈭�0鈭�X(t)dt=鈭�0鈭�E[X(t)]dt

whenever X(t),0鈮�t<鈭�,are all nonnegative random

variables; this result is indeed true. Use it to give another proof of the result that for a nonnegative random variable X,

E[X)=鈭�0鈭�P(X>t}dt

Hint: Define, for each nonnegative t, the random variable

X(t)by

role="math" localid="1647348183162" X(t)=1ift<X\\0ift鈮�X

Now relate4q

鈭�0鈭�X(t)dttoX
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

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.