Let X denote the number of bacteria in a water sample Assume

Let X denote the number of bacteria in a water sample. Assume that X is distributed Poisson with mean . Suppose that varies from location to location and has an exponential distribution with mean 10. Namely,

X ~ Poi(), and ~ Exp( = 10).

A location is randomly selected from which a water sample is taken. Find the average number of bacteria in the water sample, (i.e. find E[X]).

Note: E[X] = E[E[X|]].

Solution

Sol)

Let X follows exponential distribution with mean 10.

E[X]) =1/ =1/10 =0.1