Momentgenerating function
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.
Definition
In probability theory and statistics, the momentgenerating function of a random variable X is
wherever this expectation exists. In other terms, the momentgenerating function can be interpreted as the expectation of the random variable .
always exists and is equal to 1.
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.
More generally, where ^{T}, an ndimensional random vector, and t a fixed vector, one uses instead of tX:
The reason for defining this function is that it can be used to find all 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 hence obtain the ith moment about the origin, m_{i}, see Calculations of moments below.
Examples
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 Mx(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  
Calculation
The momentgenerating function is the expectation of a function of the random variable, it can be written as:
 For a discrete probability mass function,
 For a continuous probability density function,
 In the general case: , using the Riemann–Stieltjes integral, and where F is the cumulative distribution function.
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.
Linear combination of independent random variables
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
Vectorvalued random variables
For vectorvalued random variables X with real components, the momentgenerating function is given by
where t is a vector and is the dot product.
Important properties
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.
Calculations of moments
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.
Other properties
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:
This can be used together with Markov's inequality and Stirling's approximation to give tail bounds for positive or symmetric random variables:
Take for example the standard normal , then and so we get that , which is tight up to a factor .
Relation to other functions
Related to the momentgenerating function are a number of other transforms that are common in probability theory:
 Characteristic function
 The characteristic function is related to the momentgenerating function via the characteristic function is the momentgenerating function of iX or 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.
 Cumulantgenerating function
 The cumulantgenerating function is defined as the logarithm of the momentgenerating function; some instead define the cumulantgenerating function as the logarithm of the characteristic function, while others call this latter the second cumulantgenerating function.
 Probabilitygenerating function
 The probabilitygenerating function is defined as This immediately implies that
See also
References
 ↑ Bulmer, M.G., Principles of Statistics, Dover, 1979, pp. 75–79
 ↑ Grimmett, Geoffrey (1986). Probability  An Introduction. Oxford University Press. p. 101. ISBN 9780198532644.
 Casella, George; Berger, Roger. Statistical Inference (2nd ed.). pp. 59–68. ISBN 9780534243128.