Prove that a random variable has a Poisson distribution? If $X$ is a random variable whose PMF, $p_X(x)$, is positive on and only on non-negative integers. How can we show that $X$ has a Poisson distribution if
$$\begin{align*}
p_X(x) = \frac{3}{x} p_x(x-1) && x = 1, 2, 3, \ldots
\end{align*}$$

$$
p_X(4) = \frac 3 4 p_X(3) = \frac 3 4 \cdot \frac 3 3 p_X(2) = \frac 3 4 \cdot \frac 3 3\cdot \frac 3 2p_X(1) = \frac 3 4 \cdot \frac 3 3 \cdot \frac 3 2 \cdot\frac 3 1p_X(0) = \frac{3^4 p_X(0)}{4!}.
$$
As with $4$, so with other positive integers (or a bit less informally: prove by induction that $p_X(x)= \dfrac{3^x p_X(0)}{x!}$).
To show that this is a Poisson distribution with expected value $3$, it remains only to show that $p_X(0)=e^{-3}$.  So
$$
1=\sum_{x=0}^\infty \frac{3^x p_X(0)}{x!} = p_X(0)\sum_{x=0}^\infty \frac{3^x}{x!} = p_X(0) e^3.
$$
