The Uniqueness Property of MGFs

The uniqueness property of moment-generating functions states that if two random variables $X$ and $Y$ have MGFs that agree on some open interval around $0$, then $X$ and $Y$ have the same distribution:

$$M_X(t) = M_Y(t) \text{ for all } t \in (-\delta, \delta) \quad \Longleftrightarrow \quad X \stackrel{d}{=} Y.$$

In other words, an MGF (when it exists) is a fingerprint of a distribution. To identify the distribution of a random variable from its MGF, we pattern-match against known forms:

Distribution	MGF
$\text{Binomial}(n,p)$	$(1-p+pe^t)^n$
$\text{Poisson}(\lambda)$	$e^{\lambda(e^t-1)}$
$\text{Normal}(\mu,\sigma^2)$	$e^{\mu t + \sigma^2 t^2/2}$
$\text{Exponential}(\lambda)$	$\dfrac{\lambda}{\lambda-t}$, $t<\lambda$

For example, if $M_X(t) = e^{6(e^t - 1)}$, this matches the Poisson form with $\lambda = 6$. By the uniqueness property, $X \sim \text{Poisson}(6)$.