Moment Generating Functions

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Moment Generating Functions
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Continuous Distributions
The Uniform distribution from a to b
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The Normal distribution (mean m, standard
deviation s)
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The Exponential distribution
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Weibull distribution with parameters a and b.
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The Weibull density, f(x)
(a 0.9, b 2)
(a 0.7, b 2)
(a 0.5, b 2)
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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.
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Expectation of functions of Random Variables
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• X is discrete

X is continuous
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Moments of Random Variables
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The kth moment of X.
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• the kth central moment of X

where m m1 E(X) the first moment of X .
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Rules for expectation
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• Rules

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Moment generating functions
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• Moment Generating function of a R.V. X

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• Examples

1. The Binomial distribution (parameters p, n)

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1. The Poisson distribution (parameter l)

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

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

The moment generating function of X , mX(t) is
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We will now use the fact that
We have completed the square
This is 1
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1. The Gamma distribution (parameters a, l)

The moment generating function of X , mX(t) is
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We use the fact
Equal to 1
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Properties of Moment Generating Functions
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1. mX(0) 1

Note the moment generating functions of the
following distributions satisfy the property
mX(0) 1
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We use the expansion of the exponential function
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Now
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Property 3 is very useful in determining the
moments of a random variable X. Examples
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To find the moments we set t 0.
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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
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The moments for the standard normal distribution

• We use the expansion of eu.

We now equate the coefficients tk in
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• If k is odd mk 0.

For even 2k