Two teams have to play a final of a tournament. Team A score a number of goals that can be shaped like a random variable $X \sim Poi(\lambda_{A}=2.5)$. Team B score a number of goals that can be shaped like a random variable $Y \sim Poi(\lambda_{B}=2)$. The variables $X$ and $Y$ are independent.
1) What is the probability that A win?
2) What is the probability of a tie?
3) What is the probability of an over (that is the sum of goals of two teams is greater than $4$)?
I think I've solved only the point 2).
Knowing that $H=${total number of goals}$\sim Poi(4.5)\rightarrow \mathbb{P}(H=h)=\frac{e^{-4.5}4.5^h}{h!}$ and setting $T=X-Y$, I wrote:
$$\begin{align*} \mathbb{P}(T=0)&=\sum_{s=0}^{\infty}\mathbb{P}(X=s,Y=s)\\&=\sum_{s=0}^{\infty}\mathbb{P}(X=s,Y=s,H=2s)\\&=\sum_{s=0}^{\infty}\mathbb{P}(H=2s)\mathbb{P}(X=s,Y=s|H=2s)\\&=\sum_{s=0}^{\infty}\mathbb{P}(H=2s)\mathbb{P}(X=s|H=2s)\\&=\sum_{s=0}^{\infty}\frac{e^{-4.5}4.5^{2s}}{2s!}\binom{2s}{s}p^s(1-p)^{2s-s}\\&=\sum_{s=0}^{\infty}\frac{e^{-4.5}4.5^{2s}[p(1-p)]^s}{s!s!} \end{align*}$$
Is it correct?
Instead, I find it difficult to formalize $\mathbb{P}(X>Y)$ and $\mathbb{P}(X+Y>4)$ when $X$ and $Y$ are discrete variables. Can you help me? Thanks in advance.
For 1), given the condition you gave, you need to consider outcomes (4,0), (3,1)
For 2), again, given the condition (2,2), because you can't get a draw otherwise.
For 3), an integer $n$ can be a sum of two integers $\leq n$ in $n+1$ ways, so given independence you need the probability $$ P(X+Y=n) = \sum_{k=0}^{n}P(X=k)P(Y=n-k) $$ EDIT: for 3), also consider $X+Y=Z$, which is also Poisson with rate $\lambda_1 + \lambda_2$.