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.
The moment-generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.
Let be a random variable with CDF. The moment generating function (mgf) of (or ), denoted by , is
provided this expectation exists for in some neighborhood of 0. That is, there is an such that for all in , exists. If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist.
In other words, the moment-generating function of X is the expectation of the random variable . More generally, when , an -dimensional random vector, and is a fixed vector, one uses instead of :
always exists and is equal to 1. 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. The series expansion of is
where is the th moment. Differentiating times with respect to and setting , we obtain the th moment about the origin, ;
see Calculations of moments below.
If 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
This is consistent with the characteristic function of being a Wick rotation of when the moment generating function exists, as the characteristic function of a continuous random variable is the Fourier transform of its probability density function , and in general when a 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.
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 when the latter exists.
If random variable has moment generating function , then has moment generating function
Linear combination of independent random variables
If , 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, with M(0) = 1.
An important property of the moment-generating function is that it uniquely determines the distribution. In other words, if and are two random variables and for all values of t,
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 moment-generating function does not, because the limit
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. 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.
When is non-negative, the moment generating function gives a simple, useful bound on the moments:
For any and .
This follows from the inequality into which we can substitute implies for any .
Now, if and , this can be rearranged to .
Taking the expectation on both sides gives the bound on in terms of .
As an example, consider with degrees of freedom. Then from the examples.
Picking and substituting into the bound:
We know that in this case the correct bound is .
To compare the bounds, we can consider the asymptotics for large .
Here the moment-generating function bound is ,
where the real bound is .
The moment-generating function bound is thus very strong in this case.
The characteristic function is related to the moment-generating function via the characteristic function is the moment-generating 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.
The cumulant-generating function is defined as the logarithm of the moment-generating function; some instead define the cumulant-generating function as the logarithm of the characteristic function, while others call this latter the second cumulant-generating function.