An ecologist wants to examine the deer population in two different areas, Area $1$ and Area $2$. He assumes that the number of deer, $X$ and $Y$, in Areas $1$ and $2$ is Poisson distributed.
Question:
In an area of 10 square kilometers, the ecologist calculates that the expected number of deer is $\lambda_1 = 3$ in Area $1$ and $\lambda_2 = 2$ in Area $2$, respectively. Find $P\left(X=2\right)$ and $P\left(X\:\ge \:3\right)\:$ , and find an approximate expression for $p\left(X=Y\right)$. Specify the assumption you must use to calculate the last answer.
I know that I can use this formula: $P\left(x;\:λ\right)\:=\:\frac{\left(\left(e^{-λ}\right)\:\left(λ^x\right)\right)}{x!}$
then I can just insert $X=2$ and get the answer, but how do I do it for $P(X≥3)$? and how do I find an approximate expression for $p\left(X=Y\right)$
To calculate $\mathbb{P}[X\geq 3]$ easy observe that
$$\mathbb{P}[X\geq 3]=1-\mathbb{P}[X=0]-\mathbb{P}[X=1]-\mathbb{P}[X=2]$$
To calculate $\mathbb{P}[X=Y]$ first of all observe that
$$\mathbb{P}[X=Y]=\mathbb{P}[X=0,Y=0]+\mathbb{P}[X=1,Y=1]+\mathbb{P}[X=2,Y=2]+ \dots+$$
Now it is evident that, if you want to go on with the calculation, you have to assume independence between $X$ and $Y$ so that
$$\mathbb{P}[X=k,Y=k]=\mathbb{P}[X=k]\cdot\mathbb{P}[Y=k]$$