Moment-generating Function - Calculation

Calculation

The moment-generating function is given by the Riemann–Stieltjes integral

where F is the cumulative distribution function.

If X has a continuous probability density function ƒ(x), then MX(−t) is the two-sided Laplace transform of ƒ(x).

\begin{align} M_X(-t) & = \int_{-\infty}^\infty e^{tx} f(x)\,dx \\ & = \int_{-\infty}^\infty \left( 1+ tx + \frac{t^2x^2}{2!} + \cdots + \frac{t^nx^n}{n!} + \cdots\right) f(x)\,dx \\ & = 1 + tm_1 + \frac{t^2m_2}{2!} +\cdots + \frac{t^nm_n}{n!} +\cdots, \end{align}

where mn is the nth moment.

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