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Chapter 7: Q.7.4 (page 362)

Use the conditional variance formula to determine the variance of a geometric random variable X having parameter p.

Short Answer

Expert verified

The variance of a geometric random variable Xhaving parameter pisVar(X)=1-pp2.

Step by step solution

01

Given Information

The variance of a geometric random variable X having parameter p.

02

Explanation

Suppose a sequence of independent Bernoulli random variables Xnn鈮�1 with the parameter of success p. Let Xmark the index of first random variable which has yielded a success. We know that Zhas Geometric distribution with parameterp. Using the formula for conditional variance, we can condition on random variable X1. We have that

Var(X)=EVarX鈭�X1+VarEX鈭�X1

EVarX鈭�X1=EEX2鈭�X1-EX鈭�X12

=PX1=0路EX2鈭�X1=0-EX鈭�X1=02+ PX1=1EX2鈭�X1=1-EX鈭�X1=12

=(1-p)EX2-E(X)2+0

=(1-p)Var(X)

03

Explanation

EX鈭�X1=1路IX1=1+1+1pIX1=0

Applying the variance,

VarEX鈭�X1=p(1-p)+1+1p2p(1-p)-2p(1-p)1+1p

Var(X)=(1-p)Var(X)+1p2p(1-p)

Var(X)=1-pp2

04

Final Answer

The variance of a geometric random variable X having parameterp isVar(X)=1-pp2.

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Most popular questions from this chapter

The county hospital is located at the center of a square whose sides are 3 miles wide. If an accident occurs within this square, then the hospital sends out an ambulance. The road network is rectangular, so the travel distance from the hospital, whose coordinates are (0,0), to the point(x,y) is |x|+|y|. If an accident occurs at a point that is uniformly distributed in the square, find the expected travel distance of the ambulance.

7.2. Suppose that Xis a continuous random variable with

density function f. Show that E[IX-a鈭�]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.

A total of n balls, numbered 1through n, are put into n urns, also numbered 1through nin such a way that ball iis equally likely to go into any of the urns 1,2,..i.

Find (a) the expected number of urns that are empty.

(b) the probability that none of the urns is empty.

In the text, we noted that

E鈭�i=1鈭�Xi=鈭�i=1鈭�EXi

when the Xiare all nonnegative random variables. Since

an integral is a limit of sums, one might expect that E鈭�0鈭�X(t)dt=鈭�0鈭�E[X(t)]dt

whenever X(t),0鈮�t<鈭�,are all nonnegative random

variables; this result is indeed true. Use it to give another proof of the result that for a nonnegative random variable X,

E[X)=鈭�0鈭�P(X>t}dt

Hint: Define, for each nonnegative t, the random variable

X(t)by

role="math" localid="1647348183162" X(t)=1ift<X\\0ift鈮�X

Now relate4q

鈭�0鈭�X(t)dttoX

Consider the following dice game, as played at a certain gambling casino: Players1and 2roll a pair of dice in turn. The bank then rolls the dice to determine the outcome according to the following rule: Playeri,i=1,2,wins if his roll is strictly greater than the banks. Fori=1,2,let

Ii=1鈥呪赌呪赌呪赌�ifiwins0鈥呪赌呪赌呪赌�otherwise

and show that I1and I2are positively correlated. Explain why this result was to be expected.
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