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A mixture distribution first chooses a component $j$ with nonnegative mixture weights $w_j$ summing to one, then samples from its law $P_j$. Consequently $P(A)={\sum }_j{w}_j{P}_j(A)$, and any finite moment-generating function is the same weighted sum of the component transforms. A component may be a Dirac measure, so a mixture may have both discrete and continuous parts.

A probability distribution is moment-determinate if its moments uniquely specify it among probability distributions. A moment-generating function finite in a neighbourhood of zero is a sufficient condition. The standard normal distribution is moment-determinate.

A sum of the squares of $n$ independent random variables with the standard normal distribution has the chi-squared distribution with $n$ degrees of freedom. A scaled Gaussian integral and factorization by independence give its moment-generating function $(1-2t{)}^{-n/2}$ for $t<1/2$. For positive $n$, this expectation diverges when $t\ge 1/2$.