Show that for every n 1 the conditional distribution of x1

Question

Question: Let X₁ and X₂ be independent Poisson random variables with parameters λ₁ and λ₂.

a) Show that for every n > 1, the conditional distribution of X₁, given X₁ + X₂ = n, is binomial, and find the parameters of this binomial distribution,

b) The number of eggs laid by a certain kind of insect follows a Poisson distribution quite closely. It is known that two such insects have laid their eggs in a particular area. If the total number of eggs in the area is 150, what is the chance that the first insect laid at least 90eggs? (State your assumptions, and give approximate decimal answer.)