Use of moment generating functions

Definition Let X denote a random variable with probability density function f(x) if continuous (probability mass function p(x) if discrete) Then m X (t) = the moment generating function of X

The distribution of a random variable X is described by either 1.The density function f(x) if X continuous (probability mass function p(x) if X discrete), or 2.The cumulative distribution function F(x), or 3.The moment generating function m X (t)

Properties 1. m X (0) =

4. Let X be a random variable with moment generating function m X (t). Let Y = bX + a Then m Y (t) = m bX + a (t) = E(e [bX + a]t ) = e at m X (bt) 5. Let X and Y be two independent random variables with moment generating function m X (t) and m Y (t). Then m X+Y (t) = m X (t) m Y (t)

6. Let X and Y be two random variables with moment generating function m X (t) and m Y (t) and two distribution functions F X (x) and F Y (y) respectively. Let m X (t) = m Y (t) then F X (x) = F Y (x). This ensures that the distribution of a random variable can be identified by its moment generating function

M. G. F.’s - Continuous distributions

M. G. F.’s - Discrete distributions

Moment generating function of the gamma distribution where

using or

then

Moment generating function of the Standard Normal distribution where thus

We will use

Note: Also

Note: Also

Equating coefficients of t k, we get

Using of moment generating functions to find the distribution of functions of Random Variables

Example Suppose that X has a normal distribution with mean  and standard deviation . Find the distribution of Y = aX + b Solution: = the moment generating function of the normal distribution with mean a  + b and variance a 2  2.

Thus Z has a standard normal distribution. Special Case: the z transformation Thus Y = aX + b has a normal distribution with mean a  + b and variance a 2  2.

Example Suppose that X and Y are independent each having a normal distribution with means  X and  Y, standard deviations  X and  Y Find the distribution of S = X + Y Solution: Now

or = the moment generating function of the normal distribution with mean  X +  Y and variance Thus Y = X + Y has a normal distribution with mean  X +  Y and variance

Example Suppose that X and Y are independent each having a normal distribution with means  X and  Y, standard deviations  X and  Y Find the distribution of L = aX + bY Solution: Now

or = the moment generating function of the normal distribution with mean a  X + b  Y and variance Thus Y = aX + bY has a normal distribution with mean a  X + B  Y and variance

Special Case: Thus Y = X - Y has a normal distribution with mean  X -  Y and variance a = +1 and b = -1.

Example (Extension to n independent RV’s) Suppose that X 1, X 2, …, X n are independent each having a normal distribution with means  i, standard deviations  i (for i = 1, 2, …, n) Find the distribution of L = a 1 X 1 + a 1 X 2 + …+ a n X n Solution: Now (for i = 1, 2, …, n)

or = the moment generating function of the normal distribution with mean and variance Thus Y = a 1 X 1 + … + a n X n has a normal distribution with mean a 1  1 + …+ a n  n and variance

In this case X 1, X 2, …, X n is a sample from a normal distribution with mean , and standard deviations  and Special case:

Thus and variance has a normal distribution with mean

If x 1, x 2, …, x n is a sample from a normal distribution with mean , and standard deviations  then Summary and variance has a normal distribution with mean

Population Sampling distribution of

If x 1, x 2, …, x n is a sample from a distribution with mean , and standard deviations  then if n is large The Central Limit theorem and variance has a normal distribution with mean

We will use the following fact: Let m 1 (t), m 2 (t), … denote a sequence of moment generating functions corresponding to the sequence of distribution functions: F 1 (x), F 2 (x), … Let m(t) be a moment generating function corresponding to the distribution function F(x) then if Proof: (use moment generating functions) then

Let x 1, x 2, … denote a sequence of independent random variables coming from a distribution with moment generating function m(t) and distribution function F(x). Let S n = x 1 + x 2 + … + x n then

Is the moment generating function of the standard normal distribution Thus the limiting distribution of z is the standard normal distribution Q.E.D.