In probability theory and statistics, the moment-generating function of a real-valued 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 moment-generating functions of distributions defined by the weighted sums of random variables.
However, not all random variables have moment-generating functions.
As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0.
In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases.
[1] In other words, the moment-generating function of X is the expectation of the random variable
However, 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, 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 because it can be used to find the moments of the distribution.
is a continuous random variable, the following relation between its moment-generating function
and the two-sided Laplace transform of its probability density function
holds: since the PDF's two-sided Laplace transform is given as and the moment-generating function's definition expands (by the law of the unconscious statistician) to This is consistent with the characteristic function of
is the Fourier transform of its probability density function
is a Wick rotation of its two-sided Laplace transform in the region of convergence.
See the relation of the Fourier and Laplace transforms for further information.
has a continuous probability density function
, where the Xi are independent random variables and the ai are constants, then the probability density function for Sn is the convolution of the probability density functions of each of the Xi, and the moment-generating function for Sn is given by For vector-valued random variables
with real components, the moment-generating function is given by where
Moment generating functions are positive and log-convex,[citation needed] with M(0) = 1.
An important property of the moment-generating function is that it uniquely determines the distribution.
The log-normal distribution is an example of when this occurs.
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: That is, with n being a nonnegative integer, the nth moment about 0 is the nth derivative of the moment generating function, evaluated at t = 0.
Jensen's inequality provides a simple lower bound on the moment-generating function: where
The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X.
Various lemmas, such as Hoeffding's lemma or Bennett's inequality provide bounds on the moment-generating function in the case of a zero-mean, bounded random variable.
Taking the expectation on both sides gives the bound on
To compare the bounds, we can consider the asymptotics for large
The moment-generating function bound is thus very strong in this case.
Related to the moment-generating function are a number of other transforms that are common in probability theory: