A marine biology research group is looking for a kind of precious aquatic plant in a particular sea area. Let X be the number of the plants per cubic kilometer in the sea area and assume X has the Poisson distribution Poisson(λ). Also, the parameter λ varies with the location and has a Gamma distribution with parameters α and β. (a) What is the expected number of plants per cubic kilometer? (b) What is the variance of the number of plants per cubic kilometer? (c) What is the probability they can find at least one plant per cubic kilometer?
For 1 c)
I got up to P(X>= 1) = 1- P(X=0) but am unsure of how to compute P(X=0)