An ecologist wants to examine the deer population in two different areas, Area 1 and Area 2, at Western Canada. He assumes that the number of deer, X and Y, in Areas 1 and 2 is Poisson distributed.

(a) The ecologist reckons that the expected number of deer is respectively ₁ = 3 in area 1 and ₂ = 5 in area 2. How can I find P (X = 2) and P (X 3), and an approximate expression for P (X = Y)? Clarify the assumption you must use to calculate the last answer.

(b) If a deer is in such an area, the ecologist assumes a probability of 0.2 that it is observed. If in reality there are 5 deer in the area, what is the probability that three of them are observed? What is the expected number of deer observed?