Use%20of%20moment%20generating%20functions

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Use of moment generating functions
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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

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• 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)

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Properties

1. mX(0) 1

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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)
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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
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M. G. F.s - Continuous distributions
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M. G. F.s - Discrete distributions
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Moment generating function of the gamma
distribution
where
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using
or
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then
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Moment generating function of the Standard Normal
distribution
where
thus
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We will use
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Note
Also
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Note
Also
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Equating coefficients of tk, we get
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Using of moment generating functions to find the
distribution of functions of Random Variables
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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.
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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 .

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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
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• 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
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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
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• 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
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• Special Case

a 1 and b -1.
Thus Y X - Y has a normal distribution with
mean mX - mY and variance
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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
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• 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
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Special case

In this case X1, X2, , Xn is a sample from a
normal distribution with mean m, and standard
deviations s, and
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Thus
has a normal distribution with mean
and variance
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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
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Population
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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
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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
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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
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Is the moment generating function of the standard
normal distribution
Thus the limiting distribution of z is the
standard normal distribution
Q.E.D.