Laplace transform of a nonnegative random variable (${\psi }_X(s)$)

For a nonnegative random variable $X$, its Laplace transform is

For nonnegative $X$ with finite second moment, differentiation under the expectation gives ${\varphi }^{\prime }(\lambda )=-\mathbb{E}[X{e}^{-\lambda X}]$ for $\lambda >0$. The exponential Taylor remainder divided by ${\lambda }^2$ tends pointwise to zero and is bounded in absolute value by $X^2/2$. The dominated convergence theorem therefore proves the displayed expansion without requiring a third moment.

The Laplace transform of a nonnegative random variable determines its probability law. Indeed, $U=e^{-X}$ takes values in $(0,1]$, and $\mathbb{E}[U^n]={\psi }_X(n)$. Thus the transform determines every moment of $U$; moment determinacy on a compact interval determines the law of $U$, and the injective inverse $X=-\log U$ determines the law of $X$.