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Chapter 7: Q7.28 (page 365)

Suppose that a sequence of independent trials in which each trial is a success with probability p is performed until either a success occurs or a total of n trials has been reached. Find the mean number of trials that are performed. Hint: The computations are simplified if you use the identity that for a nonnegative integer-valued random variable X,

E[X]=鈭�i=1鈭�鈥�P{X鈮�i}

Short Answer

Expert verified

The mean number of trials that are performed1鈭�qnp.

Step by step solution

01

Given information

A sequence of independent trials in which each trial is a success with probability p is performed.

02

Solution

The probability mass function of X is

P(X=i)=qi鈭�1p鈭赌i=1,2,鈥�

Now lets consider the P(X鈮�i)=鈭�x=i鈭�鈥�P(X=x)

=鈭�x=i鈭�鈥�qx鈭�1p

=pqi鈭�1+qi+qi+1+鈥�

=pqi鈭�11+q+q2+鈥�

=pqi鈭�1(1鈭�q)鈭�1

Therefore, 1+q+q2+鈥�is an infinite geometric series

=qi鈭�1

03

Solution

Now we need to calculate

E[X]=鈭�i=1鈭�鈥�P(X鈮�i)

=鈭�i=1n鈥�P(X鈮�i)

number of trials is finite (n)

=鈭�i=1n鈥�qi鈭�1

=1+q+q2+鈥�+qn鈭�1

=1鈭�qn1鈭�q

1+q+q2+鈥�+qn-1is an finite geometric series

=1鈭�qnp

04

Final answer

The mean number of trials that are performed1鈭�qnp蹋

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Most popular questions from this chapter

Show that Y=a+bX, then

蚁(X,Y)=+1鈥呪赌呪赌呪赌�ifb>0鈭�1鈥呪赌呪赌呪赌�ifb<0

Consider n independent trials, each resulting in any one ofr possible outcomes with probabilities P1,P2,鈥�,Pr. Let X denote the number of outcomes that never occur in any of the trials. Find E[X] and show that among all probability vectors P1,鈥�,Pr,E[X] is minimized whenPi=1/r,i=1,鈥�,r.

Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability .6, compute the expected number of ducks that are hit. Assume that the number of ducks in a flock is a Poisson random variable with mean 6.

LetX have moment generating function M(t), and define唯(t)=logM(t). Show that唯''(t)t=0=Var(X).

AThere are n+1participants in a game. Each person independently is a winner with probability p. The winners share a total prize of 1 unit. (For instance, if 4people win, then each of them receives 14, whereas if there are no winners, then none of the participants receives anything.) Let A denote a specified one of the players, and let Xdenote the amount that is received by A.

(a) Compute the expected total prize shared by the players.

(b) Argue that role="math" localid="1647359898823" E[X]=1鈭�(1鈭�p)n+1n+1.

(c) Compute E[X] by conditioning on whether is a winner, and conclude that role="math" localid="1647360044853" E[(1+B)鈭�1]=1鈭�(1鈭�p)n+1(n+1)p when B is a binomial random variable with parameters n and p
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