Definition:Cumulant

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