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.
As its name implies, the moment-generating function can be used to compute a distribution’s moments: the n-th moment about 0 is the n-th 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
and the moment-generating function's definition expands (by the law of the unconscious statistician) to
is the Fourier transform of its probability density function
is of exponential order, the Fourier transform of
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.
, 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
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.
This is because in some cases, the moments exist and yet the moment-generating function does not, because the limit
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:
The moment-generating function can be used in conjunction with Markov's inequality to bound the upper tail of a real random variable X.
This statement is also called the Chernoff bound.
For example, when X is a standard normal distribution and
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.
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: