Moment Generating Functions 1/33. Contents Review of Continuous Distribution Functions 2/33.

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Presentation transcript:

Moment Generating Functions 1/33

Contents Review of Continuous Distribution Functions 2/33

Continuous Distributions The Uniform distribution from a to b

The Normal distribution (mean , standard deviation  )

The Exponential distribution

The Gamma distribution Let the continuous random variable X have density function: Then X is said to have a Gamma distribution with parameters  and.

Moment Generating function of a Random Variable X

Examples 1.The Binomial distribution (parameters p, n)

The moment generating function of X, mX(t) is: 2.The Poisson distribution (parameter ) Moment Generating function of a Random Variable X

The moment generating function of X, mX(t) is: 3.The Exponential distribution (parameter ) Moment Generating function of a Random Variable X

The moment generating function of X, mX(t) is: 4.The Standard Normal distribution (  = 0,  = 1) Moment Generating function of a Random Variable X

We will now use the fact that We have completed the square This is 1 Moment Generating function of a Random Variable X

The moment generating function of X, mX(t) is: 4.The Gamma distribution (parameters , ) Moment Generating function of a Random Variable X

We use the fact Equal to 1 Moment Generating function of a Random Variable X

Properties of Moment Generating Functions 1. m X (0) = 1 Note: the moment generating functions of the following distributions satisfy the property mX(0) = 1

We use the expansion of the exponential function: Properties of Moment Generating Functions

Now Properties of Moment Generating Functions

Property 3 is very useful in determining the moments of a random variable X. Examples Properties of Moment Generating Functions

To find the moments we set t = 0. Properties of Moment Generating Functions

The moments for the exponential distribution can be calculated in an alternative way. This is note by expanding mX(t) in powers of t and equating the coefficients of tk to the coefficients in: Properties of Moment Generating Functions

Equating the coefficients of tk we get: Properties of Moment Generating Functions

The moments for the standard normal distribution We use the expansion of e u.

The moments for the standard normal distribution We now equate the coefficients tk in:

Properties of Moment Generating Functions If k is odd:  k = 0. For even 2k:

The log of Moment Generating Functions Let l X (t) = ln m X (t) = the log of the moment generating function

The log of Moment Generating Functions

Thus l X (t) = ln m X (t) is very useful for calculating the mean and variance of a random variable

The log of Moment Generating Functions Examples 1.The Binomial distribution (parameters p, n)

The log of Moment Generating Functions

2.The Poisson distribution (parameter ) The log of Moment Generating Functions

3.The Exponential distribution (parameter ) The log of Moment Generating Functions

4.The Standard Normal distribution (  = 0,  = 1) The log of Moment Generating Functions

Summary

Expectation of functions of Random Variables X is discrete X is continuous

Moments of Random Variables The k th moment of X

Moments of Random Variables The 1 th moment of X

The k th central moment of X where  =  1 = E(X) = the first moment of X. Moments of Random Variables

Rules for expectation Rules: