momentgenerating

In probability theory, the moment generating function (MGF) of a random variable X is defined as M_X(t) = E[e^{tX}], for real numbers t for which this expectation is finite. If M_X(t) exists in a neighborhood of t = 0, the MGF encodes the moments of X: the nth moment is given by M_X^(n)(0) = E[X^n], the nth derivative of M_X at 0.

Existence and domain: The MGF exists on an open interval around 0 where E[e^{tX}] is finite. When it exists in such a neighborhood, M_X is analytic there and determines the moments of X by differentiation at 0. The MGF does not exist for all distributions; for example, the Cauchy distribution has no MGF.

Key properties: M_X(0) = 1, and M'_X(0) = E[X]. For a linear transformation Y = aX + b, M_Y(t) = e^{bt} M_X(at). If X and Y are independent, M_{X+Y}(t) = M_X(t) M_Y(t). For sums of i.i.d. variables, M_{X_1+...+X_n}(t) = M_X(t)^n.

Examples: The normal distribution N(μ, σ^2) has MGF M(t) = exp(μ t + (σ^2 t^2)/2). The Poisson distribution with parameter λ has MGF M(t) = exp(λ (e^t − 1)). The gamma distribution with shape k and scale θ has M(t) = (1 − θ t)^{−k} for t < 1/θ.

Limitations and related concepts: While MGFs are powerful for deriving moments and studying sums, not all distributions possess one. The logarithm of the MGF, the cumulant generating function, yields cumulants which linearize under sums.

See also: characteristic function, cumulants, method of moments.