Moment-generating function

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 "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.
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.
Definition.
The moment-generating function of a random variable "X" is
wherever this expectation exists. In other words, the moment-generating function of "X" is the expectation of the random variable formula_2. More generally, when formula_3, an formula_4-dimensional random vector, and formula_5 is a fixed vector, one uses formula_6 instead of formula_7:
formula_9 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 formula_10 is
Hence
where formula_13 is the formula_4th moment. Differentiating formula_15 formula_16 times with respect to formula_17 and setting formula_18, we obtain the formula_16th moment about the origin, formula_20;
see Calculations of moments below.
If formula_21 is a continuous random variable, the following relation between its moment-generating function formula_15 and the two-sided Laplace transform of its probability density function formula_23 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 formula_21 being a Wick rotation of formula_15 when the moment generating function exists, as the characteristic function of a continuous random variable formula_21 is the Fourier transform of its probability density function formula_23, and in general when a function formula_31 is of exponential order, the Fourier transform of formula_32 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.
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 formula_15 when the latter exists.
formula_34
Calculation.
The moment-generating function is the expectation of a function of the random variable, it can be written as:
Note that for the case where formula_21 has a continuous probability density function formula_31, formula_42 is the two-sided Laplace transform of formula_31.
where formula_13 is the formula_4th moment.
Linear transformations of random variables.
If random variable formula_21 has moment generating function formula_15, then formula_49 has moment generating function formula_50
Linear combination of independent random variables.
If formula_52, 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 moment-generating function for "S""n" is given by
Vector-valued random variables.
For vector-valued random variables formula_54 with real components, the moment-generating function is given by
where formula_5 is a vector and formula_57 is the dot product.
Important properties.
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 formula_21 and formula_59 are two random variables and 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 moment-generating function does not, because the limit
may not exist. The log-normal distribution is an example of when this occurs.
Calculations of moments.
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 "n"th moment about 0 is the "n"th derivative of the moment generating function, evaluated at "t" = 0.
Other properties.
Jensen's inequality provides a simple lower bound on the moment-generating function:
where formula_65 is the mean of "X".
Upper bounding 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 formula_66 is monotonically increasing for formula_67, we have
for any formula_67 and any "a", provided formula_15 exists. For example, when "X" is a standard normal distribution and formula_71, we can choose formula_72 and recall that formula_73. This gives formula_74, which is within a factor of 1+"a" of the exact value.
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.
When formula_21 is non-negative, the moment generating function gives a simple, useful bound on the moments:
For any formula_77 and formula_67.
This follows from the simple inequality formula_79 into which we can substitute formula_80 implies formula_81 for any formula_82.
Now, if formula_67 and formula_84, this can be rearranged to formula_85.
Taking the expectation on both sides gives the bound on formula_86 in terms of formula_87.
As an example, consider formula_88 with formula_89 degrees of freedom. Then we know formula_90.
Picking formula_91 and plugging into the bound, we get
We know that in this case the correct bound is formula_93.
To compare the bounds, we can consider the assymptotics for large formula_89.
Here the Mgf bound is formula_95,
where the real bound is formula_96.
The Mgf bound is thus very strong in this case.
Relation to other functions.
Related to the moment-generating function are a number of other transforms that are common in probability theory: