Solution Found!
Let X have a Poisson distribution with parameter Show that
Chapter 6, Problem 1E(choose chapter or problem)
Let X have a Poisson distribution with parameter \(\theta\). Let \(\theta\) be \(\Gamma(\alpha, \beta)\). Show that the marginal pmf of X (the compound distribution) is
\(k_{1}(x)=\frac{\Gamma(\alpha+x) \beta^{x}}{\Gamma(\alpha) x !(1+\beta)^{\alpha+x}}\), x = 0, 1, 2, 3, . . . ,
which is a generalization of the negative binomial distribution.
Questions & Answers
QUESTION:
Let X have a Poisson distribution with parameter \(\theta\). Let \(\theta\) be \(\Gamma(\alpha, \beta)\). Show that the marginal pmf of X (the compound distribution) is
\(k_{1}(x)=\frac{\Gamma(\alpha+x) \beta^{x}}{\Gamma(\alpha) x !(1+\beta)^{\alpha+x}}\), x = 0, 1, 2, 3, . . . ,
which is a generalization of the negative binomial distribution.
ANSWER:Step 1 of 3
Formula Poisson probability:
Probability density function of the gamma distribution:
