Q:

Finding parameters of a discrete random variable given its mean

Let $X$ be a non-negative random variable with
$$
P_X(n)=P(X=n)=A\frac{k^n}{n!},\quad n\in\mathbb N,\;n\geq 0.
$$
How do I find $A$ and $k$ if I know that $E[X]=a$?

A:

This is a Poisson distribution. The answer is $A=\exp(-a)$.
Notice that because $P_X$ is a probability mass function, then
\begin{eqnarray*}
  1 & = & \sum_{n = 0}^{\infty} P_X \left( n \right)\\
  & = & A \sum_{n = 0}^{\infty} \frac{k^n}{n!}\\
  & = & A \exp \left( k \right)
\end{eqnarray*}
Therefore, $A = \exp(- k)$. (here we used the series expansion of the exponential function.)
It is also known that $E[X]=k$. (the mean of the Poisson distribution is equal to its parameter.)
Finally $k=a$.

