Suppose that X represents for the number of fishes caught on a certain day at a Michigan lake,
so X only takes nonnegative integer values, i.e. X ∈ X = {0, 1, 2, . . .}. There are two situations
when X = 0 occurs: (1) X always takes zero value either for some visitors who do not fish
(denoted by G1); or (2) some visitors who did fish (denoted by G2) but did not catch any fish.
Assume that the number of fishes in G2 follow a standard Poisson distribution with parameter
λ > 0, and let θ ∈ [0, 1] be the probability that a visitor from G1. Find the pmf of X.
