Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Moment Generating Functions
Mathematically Mature: may contain mathematics beyond calculus with proofs.
Give some examples of transform methods in mathematics, science or engineering that you have seen or used and explain why transform methods are useful.
for all values for which the integral converges.
We need some tools to aid in proving theorems about random variables. In this section we develop a tool called the moment generating function which converts problems about probabilities and expectations into problems from calculus about function values and derivatives. Moment generating functions are one of the large class of transforms in mathematics that turn a diﬃcult problem in one domain into a manageable problem in another domain. Other examples are Laplace transforms, Fourier transforms, Z-transforms, generating functions, and even logarithms.
The general method can be expressed schematically in Figure 1.
Proof. To make the proof deﬁnite suppose that and are jointly continuous, with joint probability density function . Then:□
Remark. In words, the expectation of the product of independent random variables is the product of the expectations.
The moment generating function is deﬁned by
for all values for which the integral converges.
Example. The degenerate probability distribution has all the probability concentrated at a single point. That is, the degenerate random variable is a discrete random variable exhibiting certainty of outcome. If is a degenerate random variable, then with probability and is any other value with probability . The moment generating function of the degenerate random variable is particularly simple:
If the moments of order exist for , then the moment generating function is continuously diﬀerentiable up to order at . Assuming that all operations can interchanged, the moments of can be generated from by repeated diﬀerentiation:
Continuing in this way:
Remark. In words, the value of the th derivative of the moment generating function evaluated at is the value of the th moment of .
Theorem 2. If and are independent random variables with moment generating functions and respectively, then , the moment generating function of is given by . In words, the moment generating function of a sum of independent random variables is the product of the individual moment generating functions.
Proof. Using the lemma on independence above:□
Theorem 3. If the moment generating function is deﬁned in a neighborhood of then the moment generating function uniquely determines the probability distribution. That is, there is a one-to-one correspondence between the moment generating function and the distribution function of a random variable, when the moment-generating function is deﬁned and ﬁnite.
Proof. This proof is too sophisticated for the mathematical level we have now. □
By completing the square:
So returning to the calculation of the m.g.f.□
Theorem 5. If , and and and are independent, then . In words, the sum of independent normal random variables is again a normal random variable whose mean is the sum of the means, and whose variance is the sum of the variances.
Proof. We compute the moment generating function of the sum using our theorem about sums of independent random variables. Then we recognize the result as the moment generating function of the appropriate normal random variable.
An alternative visual proof that the sum of independent normal random variables is again a normal random variable using only calculus is available at The Sum of Independent Normal Random Variables is Normal.
This section is adapted from: Introduction to Probability Models, by Sheldon Ross.
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