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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]=∑i=1∞ P{X≥i}

Short Answer

Expert verified

The mean number of trials that are performed1−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−1p∀i=1,2,…

Now lets consider the P(X≥i)=∑x=i∞ P(X=x)

=∑x=i∞ qx−1p

=pqi−1+qi+qi+1+…

=pqi−11+q+q2+…

=pqi−1(1−q)−1

Therefore, 1+q+q2+…is an infinite geometric series

=qi−1

03

Solution

Now we need to calculate

E[X]=∑i=1∞ P(X≥i)

=∑i=1n P(X≥i)

number of trials is finite (n)

=∑i=1n qi−1

=1+q+q2+…+qn−1

=1−qn1−q

1+q+q2+…+qn-1is an finite geometric series

=1−qnp

04

Final answer

The mean number of trials that are performed1−qnp̣

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