Show that the probability density function of a negative

Chapter 3, Problem 133E

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QUESTION:

Problem 133E

Show that the probability density function of a negative binomial random variable equals the probability density function of a geometric random variable when r = 1. Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for a geometric random variable when r = 1.

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QUESTION:

Problem 133E

Show that the probability density function of a negative binomial random variable equals the probability density function of a geometric random variable when r = 1. Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for a geometric random variable when r = 1.

ANSWER:

Solution:

Step 1 of 2:

We have to show that the density function of a Negative binomial random variable equals the probability density function of a Geometric random variable when r = 1.

Let X follows the Negative binomial distribution then the probability density function is

f(x) =

For r = 1,  = 1, and  =

Therefore, f(x) =

Hence, f(x) =  density for Geometric distribution.


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