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Chapter 7: Q. 7.7 (page 363)

Let Xbe the smallest value obtained when knumbers are randomly chosen from the set 1,鈥�,n. Find E[X]by interpreting Xas a negative hypergeometric random variable.

Short Answer

Expert verified

The required mean is equal ton+1k+1.

Step by step solution

01

Given Information

The smallest value from the random variable is1,鈥�,n.

02

Explanation

Observe that X鈮�jmeans that we have not chosen any of value less than j, i.e., all values are greater or equal to j. The probability of that event is

P(X鈮�j)=n-j+1knk=n-kj-1nj-1

03

Explanation

So, we see that Xhas the same distribution as the random variable in Example 3e, but here we have that the total number of balls is equal to nand the number of special balls is equal to k. Therefore, we have that E(X)is equal to

E(X)=1+n-kk+1=n+1k+1

04

Final answer

The required mean is equal to n+1k+1.

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Most popular questions from this chapter

Between two distinct methods for manufacturing certain goods, the quality of goods produced by method iis a continuous random variable having distribution Fi,i=1,2. Suppose that goods are produced by method 1 and mby method 2 . Rank the n+mgoods according to quality, and let

Xj=1鈥呪赌呪赌呪赌�if thejth best was produced from鈥呪赌呪赌呪赌�method12鈥呪赌呪赌呪赌�otherwise

For the vector X1,X2,鈥�,Xn+m, which consists of n1'sand m2's, let Rdenote the number of runs of 1 . For instance, if n=5,m=2, and X=1,2,1,1,1,1,2, then R=2. If F1=F2(that is, if the two methods produce identically distributed goods), what are the mean and variance of R?

Let X1,X2,鈥�be a sequence of independent random variables having the probability mass functionPXn=0=PXn=2=1/2,n鈮�1

The random variable X=鈭�n=1鈭�Xn/3nis said to have the Cantor distribution.

Find E[X]andVar(X).

Consider an urn containing a large number of coins, and suppose that each of the coins has some probability p of turning up heads when it is flipped. However, this value of pvaries from coin to coin. Suppose that the composition of the urn is such that if a coin is selected at random from it, then the p-value of the coin can be regarded as being the value of a random variable that is uniformly distributed over 0,1. If a coin is selected at random from the urn and flipped twice, compute the probability that

a. The first flip results in a head;

b. both flips result in heads.

We say that Xis stochastically larger than Y, written X鈮�stY, if, for all t,

P{X>t}鈮�P{Y>t}

Show that if X鈮�stYthen E[X]鈮�E[Y]when

(a) Xand Yare nonnegative random variables;

(b) Xand Yare arbitrary random variables. Hint:

Write Xas

X=X+-X-

where

X+=X鈥呪赌呪赌呪赌�ifX鈮�00鈥呪赌呪赌呪赌�ifX<0,X鈭�=0ifX鈮�0鈭�XifX<0

Similarly, represent Y as Y+-Y-. Then make use of part (a).

Let X1,...be independent random variables with the common distribution functionF, and suppose they are independent of N, a geometric random variable with a parameter p. Let M=max(X1,...,XN).

(a) FindP{M鈥�x}by conditioning onN.

(b) FindP{M鈥�x|N=1}.

(c) FindP{M鈥�x|N>1}

(d) Use (b) and (c) to rederive the probability you found in (a)
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