Moment-generating function

In probability theory and statistics, the moment-generating function of a random variable is an alternative specification of its probability distribution (however, note that not all random variables have moment-generating functions). 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 moment-generating 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 general 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.

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.

$M_X(0)$ 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 absolutely. By contrast, thecharacteristic function always exists (because it is the integral of 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 tX:

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

The reason for defining this function is that it can be used to find all the moments of the distribution. The series expansion of e^tX is:

$e^{tX} = 1 + tX + \frac{t^2X^2}{2!} + \frac{t^3X^3}{3!} + \cdots +\frac{t^nX^n}{n!} + \cdots.$

Hence:

$M_X(t) = E(e^{tX}) = 1 + tm_1 + \frac{t^2m_2}{2!} + \frac{t^3m_3}{3!}+\cdots + \frac{t^nm_n}{n!}+\cdots,$

where m_n is the nth moment.

If we differentiate M_X(t) i times with respect to t and then set t = 0 we shall therefore obtain the ith moment about the origin, m_i.

Examples

Distribution	Moment-generating function M_X(t)	Characteristic function φ(t)
Bernoulli $\, P(X=1)=p$	$\, 1-p+pe^t$	$\, 1-p+pe^{it}$
Geometric $(1 - p)^{k-1}\,p\!$	$\frac{pe^t}{1-(1-p) e^t}\!$ , for $t<-\ln(1-p)\!$	$\frac{pe^{it}}{1-(1-p)\,e^{it}}\!$
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-squared 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)	$\, e^{t^\mathrm{T} \mu + \frac{1}{2} t^\mathrm{T} \Sigma t}$	$\, e^{i t^\mathrm{T} \mu - \frac{1}{2} t^\mathrm{T} \Sigma t}$
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	$\, e^{it\mu -\theta\|t\|}$
Negative Binomial NB(r, p)	$\, \frac{((1-p)e^t)^r}{(1-pe^t)^r}$	$\, \frac{((1-p)e^{it})^r}{(1-pe^{it})^r}$

So that characteristic function is a Wick rotation of the moment generating function Mx(t).

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). This statement is not equivalent to ``if two distributions have the same moments, then they are identical at all points", because in some cases the moments exist and yet the moment-generating function does not, because in some cases the limit

$\lim_{n \rightarrow \infty} \sum_{i=0}^n \frac{t^im_i}{i!}$

does not exist. This happens for the lognormal 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:

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

n should be nonnegative.

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 iXor the moment generating function of X evaluated on the imaginary axis. This function can also be viewed as the Fourier transform of the probability density function, which can therefore be deduced from it by inverse Fourier transform.

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 $G(e^t) = E[e^{tX}] = M_X(t).\,$

Monday, 21 May 2012