Moment Generating Function

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Moment Generating Function

The moment generating function of the random variable X, provided it exists, is M_x(t) = E [e^tx] where E [g(X)] denotes the expectation of the function g(X ). For example, if the random variable X follows the normal distribution with mean μ, and variance σ², the moment generating function of X is M_X (t ) = e ^{μt +σ 2}t ²/2

The moment generating function has two main uses. First, as the name implies, it can be used to obtain the moments of a random variable. Specifically, the k moment of the random variable X, α _k = E [X ^k], is given by where is the k^th derivative of M_X(t ) evaluated at t = 0. For example, if X is normally distributed with mean μ, and variance σ², and hence moment generating function M_X (t ) = e ^{µt + σ ²t ²/2}, it follows that

and

It follows that the first moment of X is μ, and the second moment of X is μ² + σ²

The second, and perhaps more important, use of the moment generating function derives from the fact that the moment generating function uniquely identifies the distribution function of a random variable. Thus, if M_X1 (t ) = M_X2 (t ) then Pr (X₁ ≤ x ) = (X₂ ≤ x ). For example, if the random variable X has the moment generating function M_X (t ) = e ^{μt + σ 2t2/2} then X necessarily follows the normal distribution. This property of the moment generating function can sometimes be used to determine the distribution of the limit of a sequence of random variables. Consider, for example, a sequence of random variables {Y_n ; n = 1, 2,…} with distribution functions {F_n (y ); n = 1, 2, …} and corresponding moment generating functions {M _n (t ); n = 1, 2 …}. If lim_n→∞ M_n (t ), where M(t ) is the moment generating function of a random variable Y with distribution function F(y ) = Pr (Y ≤ y ), then lim_{n → ∞} F_n(y ) = F(y ). F(y ) is called the limiting distribution of the sequence {Y_n ; n = 1, 2, …} and Y_n is said to converge in distribution to Y. For n sufficiently large, F(y ) provides a good approximation to the distribution of Y_n^. For example, consider the sequence of sample means {X^̄_n ; n = 1, 2, …} obtained from random samples of size n from a population with mean μ, and variance σ². Under certain conditions, the standardized sequence converges in distribution to a standard normal random variable. This result, referred to as a central limit theorem for the sample mean, is typically obtained by showing that the sequence of corresponding moment generating functions {M _{X ̄n} (t )} converges to M(t ) = e ^{t 2/2}, the moment generating function of a normal random variable with mean zero and variance 1, that is, a standard normal random variable.

Closely related to the moment generating function is the so-called characteristic function. The characteristic function of the random variable X is C_x(t ) = E [e^itX ] where and e^itX = cos(tX ) + isin(tX ). The advantage of the characteristic function is that it always exists whereas the moment generating function may not. If the moment generating function exists, the characteristic function is related to it by C_X(t ) = M_X(it ). Thus, for example, the characteristic function of a normally distributed random variable with mean μ, and variance σ² is C_X (t ) = e itµ–σ ₂t ^2/2. Like the moment generating function, the characteristic function can be used to obtain moments of the random variable. And since it uniquely identifies the distribution of the random variable, it can be used to obtain the limiting distribution of sequences of random variables.

Elementary mathematical treatments of the moment generating function are given in Donald A. Berry and Bernard W. Lindgren and John E. Freund. Intermediate treatments may be found in Robert V. Hogg and Allen T. Craig, and Lindgren and advanced mathematical discussion can be found in Harald Cramér and M. Loève. David attributes the first occurrences in print of the term moment generating function to Henri Poincaré (1912 in French) and Cecil C. Craig (1936 in English).