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Chapter 7: Q.7.14 (page 360)

For Example 2i, show that the variance of the number of coupons needed to a mass a full set is equal to∑i=1N−1 iN(N−i)2

When Nis large, this can be shown to be approximately equal (in the sense that their ratio approaches 1 as N→∞) to N2π2/6.

Short Answer

Expert verified

It has been shown that the variance of the number of coupons needed to amass a full set is equal toN→∞.

Step by step solution

01

Given Information

The variance of the number of coupons needed to a mass a full set =∑i=1N-1iN(N-i)2

Ratio approaches 1 asN→∞

02

Explanation

The variables in the summation are independent. Hence:

X=X0+X1+…+XN-1

Var(X)=∑i=1N-1VarXi

Now, to compute the required variance, one needs to use the following result:

EXi=NN-i

PXi=k=N-iNiNk-1

Using this result, one has that:

Var⁡Xi=∑k=1∞ N−iNiNk−1k2−N2(N−i)2

=N-iN∑k=1∞iNk-1k2-N2(N-i)2

=N-iNN2(N+i)(N-i)3-N2(N-i)2

03

Explanation

Simplifying the expression, one obtains that:

VarXi=N−iNN2(N+i)(N−i)3−N2(N−i)2

=N(N+i)(N-i)2-N2(N-i)2

=iN(N-i)2

=∑i=1N-1VarXi

=∑i=1N-1iN(N-i)2

04

Final Answer

Therefore, the variance of the number of coupons needed to amass a full set is equal toVarXi=∑i=1N-1iN(N-i)2.

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Most popular questions from this chapter

N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then sits either at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the N2pairs of people is, independently, a pair of friends with probability p, find the expected number of occupied tables.

Hint: Let Xiequal 1or 0, depending on whether theith arrival sits at a previously unoccupied table.

Two envelopes, each containing a check, are placed in front of you. You are to choose one of the envelopes, open it, and see the amount of the check. At this point, either you can accept that amount or you can exchange it for the check in the unopened envelope. What should you do? Is it possible to devise a strategy that does better than just accepting the first envelope? Let Aand B, A<B, denote the (unknown) amounts of the checks and note that the strategy that randomly selects an envelope and always accepts its check has an expected return of (A+B)/2. Consider the following strategy: Let F(·)be any strictly increasing (that is, continuous) distribution function. Choose an envelope randomly and open it. If the discovered check has the value x, then accept it with probability F(x)and exchange it with probability 1−F(x).

(a) Show that if you employ the latter strategy, then your expected return is greater than (A+B)/2. Hint: Condition on whether the first envelope has the value Aor B. Now consider the strategy that fixes a value x and then accepts the first check if its value is greater than x and exchanges it otherwise. (b) Show that for any x, the expected return under thex-strategy is always at least (A+B)/2and that it is strictly larger than (A+B)/2if xlies between Aand B.

(c) Let X be a continuous random variable on the whole line, and consider the following strategy: Generate the value ofX, and if X=x, then employ the x-strategy of part (b). Show that the expected return under this strategy is greater than (A+B)/2.

The number of people who enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are N floors above the ground floor, and if each person is equally likely to get off at any one of theN floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.

A coin having probability p of coming up heads is continually flipped until both heads and tails have appeared. Find

(a) the expected number of flips,

(b) the probability that the last flip lands on heads.

A coin having probability p of landing on heads is flipped n times. Compute the expected number of runs of heads of size 1 , of size 2 , and of size k,1≤k≤n.
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