The distribution of a random variable is often characterized in terms of its moment generating function (mgf), a real function whose derivatives at zero are equal to the moments of the random variable. Moment generating functions have great practical relevance not only because they can be used to easily derive moments, but also because a probability distribution is uniquely determined by its mgf, a fact that, coupled with the analytical tractability of mgfs, makes them a handy tool to solve several problems, such as deriving the distribution of a sum of two or more random variables.

It must be mentioned that not all random variables possess a moment generating function. However, all random variables possess a characteristic function, another transform that enjoys properties similar to those enjoyed by the mgf.

The following is a formal definition.

Definition
Let
be a random variable. If the expected value
exists and is finite for all real numbers
belonging to a closed interval
,
with
,
then we say that
possesses a moment generating function and the
functionis
called the **moment generating function** of
.

The next example shows how the mgf of an exponential random variable is calculated.

Example Let be a continuous random variable with supportand probability density functionwhere is a strictly positive number. The expected value can be computed as follows:Furthermore, the above expected value exists and is finite for any , provided . As a consequence, possesses a mgf:

The moment generating function takes its name by the fact that it can be used to derive the moments of , as stated in the following proposition.

Proposition If a random variable possesses a mgf , then the -th moment of , denoted by , exists and is finite for any . Furthermore,where is the -th derivative of with respect to , evaluated at the point .

Proof

Proving the above proposition is quite complicated, because a lot of analytical details must be taken care of (see e.g. Pfeiffer - 2012). The intuition, however, is straightforward. Since the expected value is a linear operator and differentiation is a linear operation, under appropriate conditions we can differentiate through the expected value:Making the substitution , we obtain

The next example shows how this proposition can be applied.

Example In the previous example we have demonstrated that the mgf of an exponential random variable isThe expected value of can be computed by taking the first derivative of the mgf:and evaluating it at :The second moment of can be computed by taking the second derivative of the mgf:and evaluating it at :And so on for higher moments.

The most important property of the mgf is the following.

Proposition (Equality of distributions) Let and be two random variables. Denote by and their distribution functions and by and their mgfs. and have the same distribution (i.e., for any ) if and only if they have the same mgfs (i.e., for any ).

Proof

For a fully general proof of this proposition see e.g. Feller (2008). We just give an informal proof for the special case in which and are discrete random variables taking only finitely many values. The "only if" part is trivial. If and have the same distribution, thenThe "if" part is proved as follows. Denote by and the supports of and and by and their probability mass functions. Denote by the union of the two supports:and by the elements of . The mgf of can be written asBy the same token, the mgf of can be written as:If and have the same mgf, then for any belonging to a closed neighborhood of zeroandRearranging terms, we obtainThis can be true for any belonging to a closed neighborhood of zero only iffor every . It follows that that the probability mass functions of and are equal. As a consequence, also their distribution functions are equal.

It must be stressed that this proposition is extremely important and relevant from a practical viewpoint: in many cases where we need to prove that two distributions are equal, it is much easier to prove equality of the moment generating functions than to prove equality of the distribution functions. Also note that equality of the distribution functions can be replaced in the proposition above by equality of the probability mass functions (if and are discrete random variables) or by equality of the probability density functions (if and are continuous random variables).

The following sections contain more details about the mgf.

Let be a random variable possessing a mgf . Definewhere are two constants and . Then, the random variable possesses a mgf and

Proof

By the very definition of mgf, we haveObviously, if is defined on a closed interval , then is defined on the interval .

Let , ..., be mutually independent random variables. Let be their sum:Then, the mgf of is the product of the mgfs of , ..., :

Proof

This is easily proved by using the definition of mgf and the properties of mutually independent variables:

Some solved exercises on moment generating functions can be found below.

Let be a discrete random variable having a Bernoulli distribution. Its support isand its probability mass function iswhere is a constant. Derive the moment generating function of , if it exists.

Solution

By the definition of moment generating function, we haveObviously, the moment generating function exists and it is well-defined because the above expected value exists for any .

Let be a random variable with moment generating functionDerive the variance of .

Solution

We can use the following formula for computing the variance:The expected value of is computed by taking the first derivative of the moment generating function:and evaluating it at :The second moment of is computed by taking the second derivative of the moment generating function:and evaluating it at :Therefore,

A random variable is said to have a Chi-square distribution with degrees of freedom if its moment generating function is defined for any and it is equal toDefine where and are two independent random variables having Chi-square distributions with and degrees of freedom respectively. Prove that has a Chi-square distribution with degrees of freedom.

Solution

The moment generating functions of and areThe moment generating function of a sum of independent random variables is just the product of their moment generating functions:Therefore, is the moment generating function of a Chi-square random variable with degrees of freedom. As a consequence, has a Chi-square distribution with degrees of freedom.

Feller, W. (2008) An introduction to probability theory and its applications, Volume 2, Wiley.

Pfeiffer, P. E. (1978) Concepts of probability theory, Dover Publications.

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