Moment-generating function: Definition from Answers.com

In probability theory and statistics, the moment-generating function of any random variable is an alternate definition of its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables.

In addition to univariate distributions, moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more generic cases.

The moment-generating function does not always exist even for real-valued arguments, unlike the characteristic function. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.

1 Definition
2 Examples
3 Calculation
- 3.1 Sum of independent random variables
- 3.2 Vector-valued random variables
4 Important properties
- 4.1 Calculations of moments
5 Relation to other functions
6 See also

Definition

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.

It should be noted that $M X (0)$ always exists and is equal to 1.

A key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge. By contrast, the characteristic function always exists (because the integral is a bounded function on a space of finite measure), and thus may be used instead.

More generally, where $\mathbf X = ( X_1, \ldots, X_n)$ , an n-dimensional random vector, one uses $\mathbf t \cdot \mathbf X = \mathbf t^\mathrm T\mathbf X$ instead of $t X$ :

$M_{\mathbf X}(\mathbf t) := E\left(e^{\mathbf t^\mathrm T\mathbf X}\right).$

Examples

Distribution	Moment-generating function M_X(t)	Characteristic function φ(t)
Binomial B(n, p)	$\, (1-p+pe^t)^n$	$\, (1-p+pe^{it})^n$
Poisson Pois(λ)	$\, e^{\lambda(e^t-1)}$	$\, e^{\lambda(e^{it}-1)}$
Uniform U(a, b)	$\, \frac{e^{tb} - e^{ta}}{t(b-a)}$	$\, \frac{e^{itb} - e^{ita}}{it(b-a)}$
Normal N(μ, σ²)	$\, e^{t\mu + \frac{1}{2}\sigma^2t^2}$	$\, e^{it\mu - \frac{1}{2}\sigma^2t^2}$
Chi-square χ²_k	$\, (1 - 2t)^{-k/2}$	$\, (1 - 2it)^{-k/2}$
Gamma Γ(k, θ)	$\, (1 - t\theta)^{-k}$	$\, (1 - it\theta)^{-k}$
Exponential Exp(λ)	$\, (1 - t\lambda^{-1})^{-1}$	$\, (1 - it\lambda^{-1})^{-1}$
Multivariate normal N(μ, Σ)	$\, e^{t'\mu + \frac{1}{2}t'\Sigma t}$	$\, e^{it'\mu - \frac{1}{2}t'\Sigma t}$
Degenerate δ_a	$\, e^{ta}$	$\, e^{ita}$
Laplace L(μ, b)	$\, \frac{e^{t\mu}}{1 - b^2t^2}$	$\, \frac{e^{it\mu}}{1 + b^2t^2}$
Cauchy Cauchy(μ, θ)	not defined	$\, e^{it\mu -\theta\|t\|}$

Calculation

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

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

where m_i is the ith moment.

Sum of independent random variables

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

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

Vector-valued random variables

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

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

where t is a vector and $\langle \cdot, \cdot \rangle$ is the dot product.

Important properties

The most important property of the moment-generating function is that if two distributions have the same moment-generating function, then they are identical at all points. That is if for all values of t,

$M_X(t) = M_Y(t),\,$

then

$F_X(x) = F_Y(x) \,$

for all values of x (or equivalently X and Y have the same distribution)

Calculations of moments

The moment-generating function is so called because, if it exists on an open interval around t = 0, then it is the exponential generating function of the moments of the probability distribution:

$E \left( X^n \right) = M_X^{(n)}(0) = \frac{d^n M_X}{dt^n}(0).$

Relation to other functions

Related to the moment-generating function are a number of other transforms that are common in probability theory:

characteristic function: The characteristic function $\varphi_X(t)$ is related to the moment-generating function via $\varphi_X(t) = M_{iX}(t) = M_X(it):$ the characteristic function is the moment-generating function of iX or the moment generating function of X evaluated on the imaginary axis.

cumulant-generating function: The cumulant-generating function is defined as the logarithm of the moment-generating function; some instead define the cumulant-generating function as the logarithm of the characteristic function, while others call this latter the second cumulant-generating function.

probability-generating function: The probability-generating function is defined as $G(z) = E[z^X].\,$ This immediately implies that $G(e^t) = E[e^{tX}] = M_X(t).\,$

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