So here's the premise: an ecologist wishes to survey the amount of deer in two areas, he assumes that the amounts of deer in the areas are Poisson distributed.
Task 1) The ecologist decides that the amount of expected deer in area 1 and 2 are respectively $\lambda=3$ in Area 1 and $\lambda=5$ in Area 2. Find $P(X=2)$ and $P(X\ge 3)$, find an approximate expression for $P(X=Y)$ and specify the prerequisite you need to calculate the answer.
I've already found the first two to be $0.22404$ and $0.57681$.
Now I don't exactly know where to go with $P(X=Y)$, $Y$ doesn't seem to be mentioned anywhere and I don't get how to use it?
Task 2: If a deer is in such an area, the ecologist assumes a probability of $0.2$ that it will be observed. If in reality there are $5$ deer in the area, what is the probability that $3$ of them are observed? What is the expected number of deer observed?
Now I have some thoughts on how I could go about this one, but I might be completely wrong. My thoughts were: $1-0.8 * 0.8 * 0.8 * 0.8 * 0.8=0.67232$ and then doing: $0.67232 * 0.67232 * 0.67232 = 0.30390$, but I guess that might be wrong?