Editing Normal distribution (section)

=== Moment- and cumulant-generating functions ===
The [[moment generating function]] of a real random variable {{tmath|X}} is the expected value of <math display=inline>e^{tX}</math>, as a function of the real parameter {{tmath|t}}. For a normal distribution with density {{tmath|f}}, mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math>, the moment generating function exists and is equal to

<math display=block>M(t) = \operatorname{E}\left[e^{tX}\right] = \hat f(it) = e^{\mu t} e^{\sigma^2 t^2/2}\,.</math>
For any {{tmath|k}}, the coefficient of {{tmath|t^k / k!}} in the moment generating function (expressed as an [[Generating function#Exponential generating function (EGF)|exponential power series]] in {{tmath|t}}) is the normal distribution's expected value {{tmath|\operatorname{E}[X^k]}}.

The [[cumulant generating function]] is the logarithm of the moment generating function, namely

<math display=block>g(t) = \ln M(t) = \mu t + \tfrac 12 \sigma^2 t^2\,.</math>

The coefficients of this exponential power series define the cumulants, but because this is a quadratic polynomial in  {{tmath|t}}, only the first two [[cumulant]]s are nonzero, namely the mean&nbsp;{{tmath|\mu}} and the variance&nbsp;{{tmath|\sigma^2}}.

Some authors prefer to instead work with the [[characteristic function (probability theory)|characteristic function]] {{math|1=E[''e''{{sup|''itX''}}] = ''e''{{sup|''iμt'' − ''σ''{{sup|2}}''t''{{sup|2}}/2}}}} and {{math|1=ln E[''e''{{sup|''itX''}}] = ''iμt'' − {{sfrac|1|2}}''σ''{{sup|2}}''t''{{sup|2}}}}.