Saturday, August 25, 2007

Moment-generating function

Moment-generating function
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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 mi is the ith moment. MX( − 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, ..., Xn 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 ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi and the moment-generating function for Sn 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.

[edit] See also

* Characteristic function
* Cumulant generating function
* Factorial moment generating function
* Rate function

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Category: Probability theory

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