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Chapter 7: Q.7.20 (page 354)

In an urn containing n balls, the ith ball has weight W(i),i = 1,...,n. The balls are removed without replacement, one at a time, according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn. For instance, if at some time i1,...,ir is the set of balls remaining in the urn, then the next selection will be ij with probability W(ij)/鈭�k=1rW(ik), j = 1,...,r Compute the expected number of balls that are withdrawn before the ball number 1is removed.

Short Answer

Expert verified

The required mean number is鈭�j=2nW(j)W(j)+W(1).

Step by step solution

01

Given Information

Given the probability that,

W(ij)/鈭�k=1rW(ik) j = 1,...,r

02

Explanation

Define indicator random variables Ij,j=2,...,n that marks if we have chosen the ball number j out from the jar before we have chosen ball number 1. The weight of the ball j is W(j) and the weight of ball 1is W(1). Observe that the probability that we have chosen ball j before ball number 1is exactly

P(Ij=1)=W(j)W(j)+W(1)

So, the expected number of balls that have been drawn before ball number 1is

鈭�j=2nP(Ij=1)=鈭�j=2nW(j)W(j)+W(1)

03

FInal Answer

The required mean number is鈭�j=2nW(j)W(j)+W(1).

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Most popular questions from this chapter

The number of accidents that a person has in a given year is a Poisson random variable with mean 位蹋 However, suppose that the value of 位changes from person to person, being equal to 2for 60percent of the population and 3for the other 40percent. If a person is chosen at random, what is the probability that he will have

(a) 0accidents and,

(b) Exactly 3accidents in a certain year? What is the conditional probability that he will have3 accidents in a given year, given that he had no accidents the preceding year?

In Example 5c, compute the variance of the length of time until the miner reaches safety.

LetU1,U2,...be a sequence of independent uniform(0,1)random variables. In Example 5i, we showed that for 0鈮�x鈮�1,E[N(x)]=ex, where

N(x)=minn:鈭�i=1n鈥�Ui>x

This problem gives another approach to establishing that result.

(a) Show by induction on n that for 0<x鈮�10 and all n鈮�0

P{N(x)鈮�n+1}=xnn!

Hint: First condition onU1and then use the induction hypothesis.

use part (a) to conclude that

E[N(x)]=ex

Suppose that X1and X2 are independent random variables having a common mean 渭. Suppose also that VarX1=蟽12 and VarX2=蟽22. The value of 渭 is unknown, and it is proposed that 渭 be estimated by a weighted average of X1 and X2. That is, 位X1+(1-位)X2 will 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.

Urn 1contains 5white and 6black balls, while urn 2contains 8white and 10black balls. Two balls are randomly selected from urn 1and are put into urn 2. If 3balls are then randomly selected from urn 2, compute the expected number of white balls in the trio.

Hint: LetXi = 1if the i th white ball initially in urn 1is one of the three selected, and let Xi = 0otherwise. Similarly, let Yi = 1if the i the white ball from urn 2is one of the three selected, and let Yi = 0otherwise. The number of white balls in the trio can now be written as鈭�15Xi+鈭�18Yi
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