Johnny Deng's Column: Moment-generating function

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.

For vector-valued random variables X with real components, the moment-generating function is given by

$M_X(\mathbf{t}) = E\left( e^{\langle \mathbf{t}, \mathbf{X}\rangle}\right)$

where t is a vector and $\langle \mathbf{t} , \mathbf{X}\rangle$ is the dot product.

Provided the moment-generating function exists in an interval around t = 0, the nth moment is given by

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

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. $M X ( - t)$ is just the two-sided Laplace transform of f(x).

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

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

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

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

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.