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Chapter 7: Q.7.18 (page 364)

Let X be the length of the initial run in a random ordering of n ones and m zeros. That is, if the first k values are the same (either all ones or all zeros), then X 脷 k. Find E[X].

Short Answer

Expert verified

The E[X}of the problem if X be the length of the initial run in a random ordering of n ones and m zeros E[L]=nm+1+mn+1.

Step by step solution

01

Given information

X= length of the initial run in a random ordering of n ones and m zeros

k-value are same

02

Solution

The solution will be shown below,

E[L]=E[L鈭�First vlaue is one]n(n+m)

+E[L鈭�First value is0]m(n+m)

Now if first value =1,

Length of run will be position of first 0when considering remaining n+m-1values, of which n-1are one's and mare zero's

鈬�E[L]=n+mm+1nn+m+n+mn+1mn+m

鈬�E[L]=nm+1+mn+1

03

Final answer

The E[X]of the given problem Will be E[L]=nm+1+mn+1.

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Most popular questions from this chapter

A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.

(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].

Hint: De铿乶e n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example 2m.

(b) Let Y denote the day on which the last large pills chosen. Find E[Y].

Hint: What is the relationship between X and Y?

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

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.

The joint density of Xand Yis given by

f(x,y)=e-yy,0<x<y,0<y<鈭�

Compute EX3鈭�Y=y.

7.4. If X and Y have joint density function fX,Y(x,y)={1/y,if0<y<1,0<x<y0,otherwisefind

(a) E[X Y]

(b) E[X]

(c) E[Y]
See all solutions

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