Expected value of the maximum of poisson distributed variable and 0

Question

I want to find the expected value of max{S-X,0}, where X is Poisson(lambda) distributed and S is a fixed number. I am looking for the notation with integrals.

Answer 1

$$ \sum_{x=0}^{\infty}\max\left(S-x,0\right)f(x) $$ where $$ f(x) = \frac{\lambda^x\mathrm{e}^{-\lambda}}{x!} $$ since the max leads to all values of $k$ greater than $S$ then the payoff is zero thus we can limit the summation $$ \sum_{x=0}^{S}\left(S-x\right)f(x) $$