How to calculate the generating function of a Poisson distribution?

Question

Hi there, I'm trying to understand the solution to the question in the title. Can anyone please explain why the final summation is equal to $e^{λs}$?

Thank you!

Answer 1

You would be presumed to have learned in calculus before studying probability that $$ e^a = \sum_{i=0}^\infty \frac{a^i}{i!}. \tag 1 $$ Without that, the function $i \mapsto \dfrac{\lambda^i}{i!} e^{-\lambda}$ for $i=0,1,2,3,\ldots$ would not be a probability mass function, since only line $(1)$ above tells you that the sum of that over all values of $i$ is $1.$

Kai-Lai Chung's undergraduate introduction to probability, in at least one of its editions, says:

"Everybody knows" that $$ e^x = \sum_{n=0}^\infty \frac{x^n}{n!}. $$

complete with quotation marks around those two words.