Essence Byrd

2022-05-01

If $X\sim $ Poisson $(\lambda )$, prove that $P[X=k]$ is maximized at $k=\lambda $

How would I go about proving something like this so that the method would work to prove a maximization point of other random variables like say Binomial is max at $k=(n+1)\star p$?

How would I go about proving something like this so that the method would work to prove a maximization point of other random variables like say Binomial is max at $k=(n+1)\star p$?

Jocelynn Meyer

Beginner2022-05-02Added 9 answers

what is $P(X=k)/P(X=k+1)$?

Is

$\frac{{e}^{\lambda}}{\lambda}$

Soo, I think you can handle it on your own

Is

$\frac{{e}^{\lambda}}{\lambda}$

Soo, I think you can handle it on your own

smachttenbem

Beginner2022-05-03Added 18 answers

The expression will depend on $k$, and it has the property that it will be less than one if and only if $P(X=k)<P(X=k+1)$

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