The Poisson pmf

Question

I have this the problem below, where I have to find the pmf.

Let $X \sim \mathcal{Poiss}(\lambda)$. Write down the pmf $p\text{x}$.

I know that a Poisson random variable has a PMF given by: $P(X = x) = \frac{\lambda^xe^{-\lambda}}{x!}$

So is this the answer to the problem, or am I missing something?

Answer 1

Yes, you have the correct equation for all nonnegative integer inputs of the PMF. However, you can improve your answer. A Poisson PMF with parameter $\lambda$ is more correctly written as: $$ P(X = k) = \begin{cases} \frac{e^{-\lambda}\lambda^{k}}{k!} & k = 0,1,2,...\\ 0 & \text{otherwise} \end{cases} $$ It is also important to note that $\lambda > 0$.