Poisson Random Variable with a Randomized Mean. The number of sales of a product in a period of length t has a Poisson probability measure pλt(n)=(λt)ne− λt / n !, n ≥ 0, with mean λt. This is true...

Poisson Random Variable with a Randomized Mean. The number of sales of a product in a period of length t has a Poisson probability measure pλt(n)=(λt)ne−^λt/ⁿ!, n ≥ 0, with mean λt. This is true when the sales over time occur according to a Poisson process with rate λ. Consider a variation of this setting in which the rate λ is a random variable Λ that may depend on the economic environment and other factors, but it is not affected by the sales. In this case, the number of sales N in the period of length t has the properties