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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,…,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,…,n.

02

Explanation

Observe that X≥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≥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

Let A1,A2,…,Anbe arbitrary events, and define

Ck={at least kof the Aioccur}. Show that

∑k=1nPCk=∑k=1nPAk

Hint: Let Xdenote the number of the Aithat occur. Show

that both sides of the preceding equation are equal to E[X].

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 μ

(b) For a fixed value of μ, what choice of P1,…,Pnmaximizes the variance of the number of successes?

(c) What choice minimizes the variance?

A prisoner is trapped in a cell containing3doors. The first door leads to a tunnel that returns him to his cell after 2days’ travel. The second leads to a tunnel that returns him to his cell after 4 days’ travel. The third door leads to freedom after 1day of travel. If it is assumed that the prisoner will always select doors 1,2, and 3 with respective probabilities .5,.3, and .2, what is the expected number of days until the prisoner reaches freedom?

Let X1,X2,…be a sequence of independent and identically distributed continuous random variables. Let N≥2be such that

X1≥X2≥⋯≥XN-1<XN

That is, Nis the point at which the sequence stops decreasing. Show that E[N]=e.

Hint: First find P{N≥n}.

7.2. Suppose that Xis a continuous random variable with

density function f. Show that E[IX-a∣]is minimized

when ais equal to the median of F.

Hint: Write

E[IX-al]=|x-a|f(x)dx

Now break up the integral into the regions where x<a

and where x>a, and differentiate.
See all solutions

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