Use of moment generating functions 1.Using the moment generating functions of X, Y, Z, …determine the moment generating function of W = h(X, Y, Z, …). 2.Identify the distribution of W from its moment generating function This procedure works well for sums, linear combinations etc.

Therorem Let X and Y denote a independent random variables each having a gamma distribution with parameters (,  1 ) and (,  2 ). Then W = X + Y has a gamma distribution with parameters (,  1 +  2 ). Proof:

Recognizing that this is the moment generating function of the gamma distribution with parameters (,  1 +  2 ) we conclude that W = X + Y has a gamma distribution with parameters (,  1 +  2 ).

Therorem (extension to n RV’s) Let x 1, x 2, …, x n denote n independent random variables each having a gamma distribution with parameters (,  i ), i = 1, 2, …, n. Then W = x 1 + x 2 + … + x n has a gamma distribution with parameters (,  1 +  2 +… +  n ). Proof:

Recognizing that this is the moment generating function of the gamma distribution with parameters (,  1 +  2 +…+  n ) we conclude that W = x 1 + x 2 + … + x n has a gamma distribution with parameters (,  1 +  2 +…+  n ). Therefore

Therorem Suppose that x is a random variable having a gamma distribution with parameters (,  ). Then W = ax has a gamma distribution with parameters ( /a,  ). Proof:

1.Let X and Y be independent random variables having a  2 distribution with 1 and 2 degrees of freedom respectively then X + Y has a  2 distribution with degrees of freedom Special Cases 2.Let x 1, x 2,…, x n, be independent random variables having a  2 distribution with 1, 2,…, n degrees of freedom respectively then x 1 + x 2 +…+ x n has a  2 distribution with degrees of freedom 1 +…+ n. Both of these properties follow from the fact that a  2 random variable with degrees of freedom is a  random variable with  = ½ and  = /2.

If z has a Standard Normal distribution then z 2 has a  2 distribution with 1 degree of freedom. Recall Thus if z 1, z 2,…, z are independent random variables each having Standard Normal distribution then has a  2 distribution with degrees of freedom.

Therorem Suppose that U 1 and U 2 are independent random variables and that U = U 1 + U 2 Suppose that U 1 and U have a  2 distribution with degrees of freedom 1 and respectively. ( 1 < ) Then U 2 has a  2 distribution with degrees of freedom 2 = - 1 Proof:

Q.E.D.

Distribution of the sample variance

Properties of the sample variance Proof:

Special Cases 1.Setting a = 0. Computing formula

2.Setting a = .

Distribution of the sample variance Let x 1, x 2, …, x n denote a sample from the normal distribution with mean  and variance  2. Let Then has a  2 distribution with n degrees of freedom.

Note: or U = U 2 + U 1 has a  2 distribution with n degrees of freedom.

has normal distribution with mean  and variance  2 /n Thus has a  2 distribution with 1 degree of freedom. We also know that has a Standard Normal distribution and

If we can show that U 1 and U 2 are independent then has a  2 distribution with n - 1 degrees of freedom. The final task would be to show that are independent

Summary Let x 1, x 2, …, x n denote a sample from the normal distribution with mean  and variance  has a  2 distribution with = n - 1 degrees of freedom. 1.than has normal distribution with mean  and variance  2 /n

The Transformation Method Theorem Let X denote a random variable with probability density function f(x) and U = h(X). Assume that h(x) is either strictly increasing (or decreasing) then the probability density of U is:

Proof Use the distribution function method. Step 1 Find the distribution function, G(u) Step 2 Differentiate G (u ) to find the probability density function g(u)

hence

or

Example Suppose that X has a Normal distribution with mean  and variance  2. Find the distribution of U = h(x) = e X. Solution:

hence This distribution is called the log-normal distribution

log-normal distribution

The Transfomation Method (many variables) Theorem Let x 1, x 2,…, x n denote random variables with joint probability density function f(x 1, x 2,…, x n ) Let u 1 = h 1 (x 1, x 2,…, x n ). u 2 = h 2 (x 1, x 2,…, x n ). u n = h n (x 1, x 2,…, x n ).  define an invertible transformation from the x’s to the u’s

Then the joint probability density function of u 1, u 2,…, u n is given by: where Jacobian of the transformation

Example Suppose that x 1, x 2 are independent with density functions f 1 (x 1 ) and f 2 (x 2 ) Find the distribution of u 1 = x 1 + x 2 u 2 = x 1 - x 2 Solving for x 1 and x 2 we get the inverse transformation

The Jacobian of the transformation

The joint density of x 1, x 2 is f(x 1, x 2 ) = f 1 (x 1 ) f 2 (x 2 ) Hence the joint density of u 1 and u 2 is:

From We can determine the distribution of u 1 = x 1 + x 2

Hence This is called the convolution of the two densities f 1 and f 2.

Example: The ex-Gaussian distribution 1. X has an exponential distribution with parameter. 2. Y has a normal (Gaussian) distribution with mean  and standard deviation . Let X and Y be two independent random variables such that: Find the distribution of U = X + Y. This distribution is used in psychology as a model for response time to perform a task.

Now The density of U = X + Y is :.

or

Where V has a Normal distribution with mean and variance  2. Hence Where  (z) is the cdf of the standard Normal distribution

g(u)g(u) The ex-Gaussian distribution