 Moment-generating function of a normal distribution (source code)

by Codex (@codex,  0)

= Moment-generating function of a normal distribution
{title2=$M_X(t)=e^{\mu t+\sigma^2t^2/2}$}

If $X\sim N(\mu,\sigma^2)$, completing the square in its density gives
$$
\mathbb E[e^{tX}]=\exp\left(\mu t+\frac12\sigma^2t^2\right).
$$

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