Moment Generating Functions

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Presentation transcript:

Moment Generating Functions

Continuous Distributions The Uniform distribution from a to b

The Normal distribution (mean m, standard deviation s)

The Exponential distribution

Weibull distribution with parameters a and b.

The Weibull density, f(x) (a = 0.9, b = 2) (a = 0.7, b = 2) (a = 0.5, b = 2)

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

Expectation of functions of Random Variables

X is discrete X is continuous

Moments of Random Variables

The kth moment of X.

the kth central moment of X where m = m1 = E(X) = the first moment of X .

Rules for expectation

Rules:

Moment generating functions

Moment Generating function of a R.V. X

Examples The Binomial distribution (parameters p, n)

The Poisson distribution (parameter l) The moment generating function of X , mX(t) is:

The Exponential distribution (parameter l) The moment generating function of X , mX(t) is:

The Standard Normal distribution (m = 0, s = 1) The moment generating function of X , mX(t) is:

We will now use the fact that We have completed the square This is 1

The Gamma distribution (parameters a, l) The moment generating function of X , mX(t) is:

We use the fact Equal to 1

Properties of Moment Generating Functions

mX(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:

Now

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

To find the moments we set t = 0.

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: Equating the coefficients of tk we get:

The moments for the standard normal distribution We use the expansion of eu. We now equate the coefficients tk in:

If k is odd: mk = 0. For even 2k: