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Title: Use%20of%20moment%20generating%20functions


1
Use of moment generating functions
2
Definition
  • Let X denote a random variable with probability
    density function f(x) if continuous (probability
    mass function p(x) if discrete)
  • Then
  • mX(t) the moment generating function of X

3
  • The distribution of a random variable X is
    described by either
  • The density function f(x) if X continuous
    (probability mass function p(x) if X discrete),
    or
  • The cumulative distribution function F(x), or
  • The moment generating function mX(t)

4
Properties
  1. mX(0) 1

5
  1. Let X be a random variable with moment
    generating function mX(t). Let Y bX a

Then mY(t) mbX a(t) E(e bX at)
eatmX (bt)
  1. Let X and Y be two independent random variables
    with moment generating function mX(t) and mY(t) .

Then mXY(t) mX (t) mY (t)
6
  1. Let X and Y be two random variables with moment
    generating function mX(t) and mY(t) and two
    distribution functions FX(x) and FY(y)
    respectively.

Let mX (t) mY (t) then FX(x) FY(x).
This ensures that the distribution of a random
variable can be identified by its moment
generating function
7
M. G. F.s - Continuous distributions
8
M. G. F.s - Discrete distributions
9
Moment generating function of the gamma
distribution
where
10
using
or
11
then
12
Moment generating function of the Standard Normal
distribution
where
thus
13
We will use
14
Note
Also
15
Note
Also
16
Equating coefficients of tk, we get
17
Using of moment generating functions to find the
distribution of functions of Random Variables
18
Example
  • Suppose that X has a normal distribution with
    mean m and standard deviation s.
  • Find the distribution of Y aX b

Solution
the moment generating function of the normal
distribution with mean am b and variance a2s2.
19
Thus Y aX b has a normal distribution with
mean am b and variance a2s2.
Special Case the z transformation
  • Thus Z has a standard normal distribution .

20
Example
  • Suppose that X and Y are independent each having
    a normal distribution with means mX and mY ,
    standard deviations sX and sY
  • Find the distribution of S X Y

Solution
Now
21
  • or

the moment generating function of the normal
distribution with mean mX mY and variance
Thus Y X Y has a normal distribution with
mean mX mY and variance
22
Example
  • Suppose that X and Y are independent each having
    a normal distribution with means mX and mY ,
    standard deviations sX and sY
  • Find the distribution of L aX bY

Solution
Now
23
  • or

the moment generating function of the normal
distribution with mean amX bmY and variance
Thus Y aX bY has a normal distribution with
mean amX BmY and variance
24
  • Special Case

a 1 and b -1.
Thus Y X - Y has a normal distribution with
mean mX - mY and variance
25
Example (Extension to n independent RVs)
  • Suppose that X1, X2, , Xn are independent each
    having a normal distribution with means mi,
    standard deviations si (for i 1, 2, , n)
  • Find the distribution of L a1X1 a1X2 anXn

Solution
(for i 1, 2, , n)
Now
26
  • or

the moment generating function of the normal
distribution with mean and variance
Thus Y a1X1 anXn has a normal
distribution with mean a1m1 anmn and
variance
27
Special case

In this case X1, X2, , Xn is a sample from a
normal distribution with mean m, and standard
deviations s, and
28
Thus
has a normal distribution with mean
and variance
29
Summary
If x1, x2, , xn is a sample from a normal
distribution with mean m, and standard deviations
s, then

has a normal distribution with mean
and variance
30
Population
31
The Central Limit theorem
If x1, x2, , xn is a sample from a distribution
with mean m, and standard deviations s, then if n
is large

has a normal distribution with mean
and variance
32
Proof (use moment generating functions)
We will use the following fact Let m1(t),
m2(t), denote a sequence of moment generating
functions corresponding to the sequence of
distribution functions F1(x) , F2(x), Let
m(t) be a moment generating function
corresponding to the distribution function F(x)
then if

then
33
Let x1, x2, denote a sequence of independent
random variables coming from a distribution with
moment generating function m(t) and distribution
function F(x).

Let Sn x1 x2 xn then
34

35

36

37

38

Is the moment generating function of the standard
normal distribution
Thus the limiting distribution of z is the
standard normal distribution
Q.E.D.
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