Moment-generating function

In probability theory and statistics, the moment-generating function of a random variable X is

$M_X(t)=E\left(e^{tX}\right).$

The moment-generating function generates the moments of the probability distribution, as follows:

$E\left(X^n\right)=M_X^{(n)}(0)=\left.\frac{\mathrm{d}^n}{\mathrm{d}t^n}\right|_{t=0} M_X(t).$

If X has a continuous probability density function f(x) then the moment generating function is given by

$M_X(t) = \int_{-\infty}^\infty e^{tx} f(x)\,\mathrm{d}x$

$= \int_{-\infty}^\infty \left( 1+ tx + \frac{t^2x^2}{2!} + \cdots\right) f(x)\,\mathrm{d}x$

$= 1 + tm_1 + \frac{t^2m_2}{2!} +\cdots,$

where m_i is the ith moment.

Regardless of whether probability distribution is continuous or not, the moment-generating function is given by the Riemann-Stieltjes integral

$\int_{-\infty}^\infty e^{tx}\,dF(x)$

Related concepts include the characteristic function, the probability-generating function, and the cumulant-generating function. The cumulant-generating function is the logarithm of the moment-generating function.

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