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Chapter 7: Q. 7.8 (page 364)

An arriving plane carries r families. A total of njof these families have checked in a total of jpieces of luggage, 鈭�inj=r. Suppose that when the plane lands, the N=鈭�ijnjpieces of luggage come out of the plane in a random order. As soon as a family collects all of its luggage, it immediately departs the airport. If the Sanchez family checked in j pieces of luggage, find the expected number of families that depart after they do.

Short Answer

Expert verified

The expected value is equal to鈭�ini路ii+j.

Step by step solution

01

Given Information

An arriving plane carries r families as鈭�inj=r.

02

Explanation

The number of families that depart after Sanchez family can be described as X=鈭�k=1r-1Ik

Observe that kth family leaves later if and only if all Sanchez's luggages have been placed before the last luggage of kth family. So, we can only consider i+jluggages and the probability that the last luggage comes from the set that has imembers is equal to

PIk=1=ii+j

03

Explanation

Now, we group random variables Ikaccording to the number of luggages they posses. We have

X=鈭�i鈭�ki=1niIki

Therefore, E(X)=鈭�i鈭�ki=1niEIki=鈭�i鈭�ki=1niPIki=1

=鈭�ini路ii+j.

04

Final answer

The expected value is equal to鈭�ini路ii+j.

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Most popular questions from this chapter

Let X1,X2,鈥�,Xn be independent and identically distributed positive random variables. For k鈮�n findE鈭�i=1kXi鈭�i=1nXi.

A group of nmen and n women is lined up at random.

(a) Find the expected number of men who have a woman next to them.

(b) Repeat part (a), but now assuming that the group is randomly seated at a round table.

A deck of n cards numbered 1 through n is thoroughly shuf铿俥d so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.

(a) If you are not given any information about your earlier guesses, show that for any strategy, E[N]=1.

(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that under this strategy

E[N]=1n+1n鈭�1+鈰�+1鈮�鈭�1n1xdx=logn

(c) Supposethatyouaretoldaftereachguesswhetheryou are right or wrong. In this case, it can be shown that the strategy that maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that

E[N]=1+12!+13!+鈰�+1n!鈮�e鈭�1

Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables.

If E[X]=1and Var(X)=5find

(a)E[(2+X2)]

(b)Var(4+3X)

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.
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