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Show that the probability density function of a negative
Chapter 3, Problem 133E(choose chapter or problem)
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.
Questions & Answers
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.