a field cantains two types of plants. type A is attractive to insects, whereas B is not. The number of insects to be found on a plant of type A can be modelled as a poisson random variable with mean m. no insects will ever be found on a plant B. the proportion of type B plants in the field is p. let X be the number of insects found on a randomly chosen plant.
1)find the probability mass function of X
2)show that the probability generating function of X is given by
π(s)=p+(1-p)*(e^(m*(s-1)). use this result to find the mean and variance of X.
(since X = 0 always for plant B and sometimes for Plant A),
(since X is never greater than zero for Plant B but can be for Plant A).
The sum is just the probability generating function of a Poisson random variable and is well known (and is probably derived somewhere in these forums).
You should know how to get mean and variance from a probability generating function.