Moment-generating function

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

$M_X(t)=E\left(e^{tX}\right), \quad t \in \mathbb{R},$

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution, as follows. Provided the moment-generating function exists in an interval around t = 0,

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

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

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

where F is the cumulative distribution function.

If X₁, X₂, ..., X_n is a sequence of independent (and not necessarily identically distributed) random variables, and

$S_n = \sum_{i=1}^n a_i X_i,$

where the a _i are constants, then the probability density function for S _n is the convolution of the probability density functions of each of the X _i and the moment-generating function for S _n is given by

$M_{S_n}(t)=M_{X_1}(a_1t)M_{X_2}(a_2t)\ldots M_{X_n}(a_nt).$

Related to the moment-generating function are a number of other transforms that are common in probability theory, including the characteristic function and the probability-generating function.

The cumulant-generating function is the logarithm of the moment-generating function.

Categories: Probability theory

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01-04-2007 01:21:04

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