The number of fishes caught by a skilled fisherman each day has a Poisson distribution with mean $2.5$. After $50$ days, what is the mean and variance for the total number of fishes the fisherman would have caught ? I know that the $$p(x=k)=\dfrac{e^{-2.5}2.5^{k}}{k!}$$ Now I can't really figure out what next to do. Will the total number of fishes be $50(2.5)$ ?
Would it be right to say that $$E(X)=\sum_{x=0}^{20}x\dfrac{e^{-2.5}2.5^{x}}{x!}$$ and $$V(X)=\left(\sum_{x=0}^{50}x^{2}\dfrac{e^{-2.5}2.5^{x}}{x!}\right)-(E(X))^{2}$$