maximum values of the Poisson distribution

Question

good day Can anybody help me? thanks

let $X\sim \operatorname{Poisson}(\lambda)$

1.- For what $\lambda$ values the value of $P (X = i), i≥0$ is maximum

2.- Prove that: $E (X^n) = \lambda E[(X+1)^{n-1} ]$ and use this result to calculate the value of $E (X^3)$

Answer 1

Hint: (1) Do you know the formula for $\mathbb P(X=i)$? Can you differentiate it with respect to $\lambda$?

(2) write the expectations using the formula for $\mathbb P(X=i)$ and compare the term for $i$ on one side to the term for $i-1$ on the other.