Moment-generating function

The moment-generating function of a real random variable $X$ is the exponential transformallowing the value $+\infty$. It always satisfies $M_X(0)=1$. The important regularity condition is finiteness on an open neighbourhood of zero; finiteness only at zero provides no moment information.

Under that neighbourhood condition, differentiating under the expected value gives $M_X^{(k)}(0)=\mathbb{E}{X}^k$ for every nonnegative integer $k$. In particular the mean is $M_X^{\prime }(0)$ and the variance is $M_X^{\prime \prime }(0)-M_X^{\prime }(0{)}^2$. Exponential bounds dominate powers locally and justify the derivatives. For independent random variables, $M_{X+Y}(t)=M_X(t)M_Y(t)$ wherever finite, making addition easier to analyze.

For a Bernoulli random variable with success probability $p$, $M_X(t)=1-p+p{e}^t$. For an exponential distribution of rate $\lambda >0$, $M_X(t)=\lambda /(\lambda -t)$ for $t<\lambda$, and it is infinite for $t\ge \lambda$. A standard Cauchy random variable has infinite $M_X(t)$ at every nonzero real $t$, although its characteristic function $\mathbb{E}{e}^{itX}$ is defined for every real $t$.

Two probability distributions with equal moment-generating functions on a neighbourhood of zero are equal. The neighbourhood condition is what makes this uniqueness theorem stronger than merely listing formal moments.