Definition:Cumulant
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Definition
Let $\map M t$ be the moment generating function of a random variable.
Let $\kappa_r$ be the coefficient of $\dfrac {t^r} {r!}$ in the power series expansion of $\ln \map M t$.
$\kappa_r$ is known as the $r$th cumulant of $\map M t$.
Examples
Expectation
The first cumulant $\kappa_1$ of a moment generating function is the expectation.
Variance
The second cumulant $\kappa_2$ of a moment generating function is the variance.
Also see
- Results about cumulants can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cumulant
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cumulant