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Moment Generating Functions

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Title: Moment Generating Functions


1
Moment Generating Functions
2
Continuous Distributions
The Uniform distribution from a to b
3
The Normal distribution (mean m, standard
deviation s)
4
The Exponential distribution
5
Weibull distribution with parameters a and b.
6
The Weibull density, f(x)
(a 0.9, b 2)
(a 0.7, b 2)
(a 0.5, b 2)
7
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.
8
Expectation of functions of Random Variables
9
  • X is discrete

X is continuous
10
Moments of Random Variables
11
The kth moment of X.
12
  • the kth central moment of X

where m m1 E(X) the first moment of X .
13
Rules for expectation
14
  • Rules

15
Moment generating functions
16
  • Moment Generating function of a R.V. X

17
  • Examples
  1. The Binomial distribution (parameters p, n)

18
  1. The Poisson distribution (parameter l)

The moment generating function of X , mX(t) is
19
  1. The Exponential distribution (parameter l)

The moment generating function of X , mX(t) is
20
  1. The Standard Normal distribution (m 0, s 1)

The moment generating function of X , mX(t) is
21
We will now use the fact that
We have completed the square
This is 1
22
  1. The Gamma distribution (parameters a, l)

The moment generating function of X , mX(t) is
23
We use the fact
Equal to 1
24
Properties of Moment Generating Functions
25
  1. mX(0) 1

Note the moment generating functions of the
following distributions satisfy the property
mX(0) 1
26
We use the expansion of the exponential function
27
Now
28
Property 3 is very useful in determining the
moments of a random variable X. Examples
29
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30
To find the moments we set t 0.
31
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33
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
34
The moments for the standard normal distribution
  • We use the expansion of eu.

We now equate the coefficients tk in
35
  • If k is odd mk 0.

For even 2k
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