In probability theory and statistics, the momentgenerating function of a realvalued random variable is an alternative specification 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 momentgenerating functions of distributions defined by the weighted sums of random variables. Note, however, that not all random variables have momentgenerating functions.
In addition to realvalued distributions (univariate distributions), momentgenerating functions can be defined for vector or matrixvalued random variables, and can even be extended to more general cases.
The momentgenerating function of a realvalued distribution does not always exist, unlike the characteristic function. There are relations between the behavior of the momentgenerating function of a distribution and properties of the distribution, such as the existence of moments.
The momentgenerating function of a random variable X is
wherever this expectation exists. In other words, the momentgenerating function is the expectation of the random variable . More generally, when ^{T}, an ndimensional random vector, and t is a fixed vector, one uses instead of tX:
always exists and is equal to 1. However, a key problem with momentgenerating functions is that moments and the momentgenerating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead.
The moment generating function is so named as it can be used to find the moments of the distribution.^{[1]} The series expansion of e^{tX} is:
Hence:
where m_{n} is the nth moment. Differentiating M_{X}(t) i times with respect to t and setting t = 0, we obtain the ith moment about the origin, m_{i}; see Calculations of moments below.
If X is a continuous random variable, the following relation between its moment generating function M_{X}(t) and the twosided Laplace transform of its probability density function f_{X}(x) holds:
as the PDF's twosided Laplace transform is given as
and the moment generating function's definition expands to
This is consistent with the characteristic function of X being a Wick rotation of M_{X}(t) when the moment generating function exists, as the characteristic function of a continuous random variable X is the Fourier transform of its probability density function f_{X}(x), and in general when a function f(x) is of exponential order, the Fourier transform of f is a Wick rotation of its twosided Laplace transform in the region of convergence. See the relation of the Fourier and Laplace transforms for further information.
Here are some examples of the moment generating function and the characteristic function for comparison. It can be seen that the characteristic function is a Wick rotation of the moment generating function M_{X}(t) when the latter exists.
Distribution  Momentgenerating function M_{X}(t)  Characteristic function φ(t) 

Bernoulli  
Geometric  

Binomial B(n, p)  
Poisson Pois(λ)  
Uniform (continuous) U(a, b)  
Uniform (discrete) U(a, b)  
Normal N(μ, σ^{2})  
Chisquared χ^{2}_{k}  
Gamma Γ(k, θ)  ;  
Exponential Exp(λ)  
Multivariate normal N(μ, Σ)  
Degenerate δ_{a}  
Laplace L(μ, b)  
Negative Binomial NB(r, p)  
Cauchy Cauchy(μ, θ)  Does not exist 
The momentgenerating function is the expectation of a function of the random variable, it can be written as:
Note that for the case where X has a continuous probability density function ƒ(x), M_{X}(−t) is the twosided Laplace transform of ƒ(x).
where m_{n} is the nth moment.
If , where the X_{i} are independent random variables and 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 momentgenerating function for S_{n} is given by
For vectorvalued random variables X with real components, the momentgenerating function is given by
where t is a vector and is the dot product.
Moment generating functions are positive and logconvex, with M(0) = 1.
An important property of the momentgenerating function is that if two distributions have the same momentgenerating function, then they are identical at almost all points.^{[2]} That is, if for all values of t,
then
for all values of x (or equivalently X and Y have the same distribution). This statement is not equivalent to the statement "if two distributions have the same moments, then they are identical at all points." This is because in some cases, the moments exist and yet the momentgenerating function does not, because the limit
may not exist. The lognormal distribution is an example of when this occurs.
The momentgenerating 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:
Here n must be a nonnegative integer.
Jensen's inequality provides a simple lower bound on the momentgenerating function:
where is the mean of X.
Hoeffding's lemma provides a bound on the momentgenerating function in the case of a zeromean, bounded random variable.
When all moments are nonnegative, the moment generating function gives a simple, useful bound on the moments:
The momentgenerating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X. Since is monotonically increasing for , we have
for any and any a, provided exists. For example, when X is a standard normal distribution and , we can choose and recall that . This gives , which is within a factor of 1+a of the exact value.
Related to the momentgenerating function are a number of other transforms that are common in probability theory:
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