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Chapter 7: Q.7.30 (page 365)

In the generalized match problem, there are n individuals of whom ni wear hat size i,鈭�i=1r鈥�ni=n. There are also n hats, of which hi are of sizei,鈭�i=1r鈥�hi=n. If each individual randomly chooses a hat (without replacement), find the expected number who choose a hat that is their size

Short Answer

Expert verified

The expected number that chooses a hat that is their size is will beE[X]=1n鈭�i=1r鈥�hini.

Step by step solution

01

Given information

The available information is. There are n individuals, ni of them wear a hat of size i. Also, there are n hats, of which hi are of size i.

02

Solution

We need to find the new variable,

Xi,j=1鈥呪赌呪赌呪赌�if thejth individual choose hat size ofi0鈥呪赌呪赌呪赌�if not

The number of individuals that choose a hat of there is given by,

X=鈭�i=1r鈥�鈭�j=1ni鈥�Xi,j

The probability of choosing a hat of size hiis hin

Thus the expected value of X will be,

E[X]=E鈭�i=1r鈥�鈭�j=1nj鈥�Xi,j

=鈭�i=1r鈥�鈭�j=1nj鈥�EXi,j

=鈭�i=1r鈥�鈭�j=1ni鈥�hin

=1n鈭�i=1r鈥�hi脳ni

=1n鈭�i=1r鈥�hini

03

Final answer

The expected number that chooses a hat that is their size is will beE[X]=1n鈭�i=1r鈥�hini

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Most popular questions from this chapter

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain

(a) 2 aces;

(b) 5 spades;

(c) all 13 hearts.

The best quadratic predictor ofYwith respect to Xis a + bX+cX2, where a, b, and c are chosen to minimize E[(Y鈭�(a+bX+cX2))2]. Determine a, b, and c.

The random variables X and Y have a joint density function is given by

f(x,y)={2e鈭�2x/x0鈮�x<鈭�,0鈮�y鈮�x0otherwise

ComputeCov(X,Y)

Consider nindependent trials, the ithof which results in a success with probability Pl.

(a) Compute the expected number of successes in the ntrials-call it 渭

(b) For a fixed value of 渭, what choice of P1,鈥�,Pnmaximizes the variance of the number of successes?

(c) What choice minimizes the variance?

In Problem 7.9, compute the variance of the number of empty urns.
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