Chapter 5.11, Problem 136E

In Exercise 5.42, the number of defects per yard in a certain fabric, Y, was known to have a Poisson distribution with parameter λ. The parameter λ was assumed to be a random variable with a density function given by

$f(\text{λ})=\{\begin{array}{l}{e}_{}^{-\text{λ}},\text{λ}\ge 0,\hfill \\ 0,\text{elsewhere}\text{.}\hfill \end{array}$

a Find the expected number of defects per yard by first finding the conditional expectation of Y for given λ.

b Find the variance of Y.

c Is it likely that Y exceeds 9?

5.42 The number of defects per yard Y for a certain fabric is known to have a Poisson distribution with parameter λ. However, λ itself is a random variable with probability density function given by

$f(\text{λ})=\{\begin{array}{l}{e}_{}^{-\text{λ}},\text{λ}\ge 0,\hfill \\ 0,\text{elsewhere}\text{.}\hfill \end{array}$

Find the unconditional probability function for Y.