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

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definition
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

slide3
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)
properties
Properties
  • mX(0) = 1
slide5

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 + a]t) = eatmX (bt)

  • Let X and Y be two independent random variables with moment generating function mX(t) and mY(t) .

Then mX+Y(t) = mX (t) mY (t)

slide6

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

slide14

Note:

Also

slide15

Note:

Also

using of moment generating functions to find the distribution of functions of random variables

Using of moment generating functions to find the distribution of functions of Random Variables

example
Example

Suppose that X has a normal distribution with mean mand 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.

slide19

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 .

example20
Example

Suppose that X and Y are independent eachhaving a normal distribution with means mX and mY , standard deviations sX and sY

Find the distribution of S = X + Y

Solution:

Now

slide21
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

example22
Example

Suppose that X and Y are independent eachhaving a normal distribution with means mX and mY , standard deviations sX and sY

Find the distribution of L = aX + bY

Solution:

Now

slide23
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

slide24

a = +1 and b = -1.

Special Case:

Thus Y = X - Y has a normal distribution with mean mX - mY and variance

example extension to n independent rv s
Example (Extension to n independent RV’s)

Suppose that X1, X2, …, Xn are independent eachhaving 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

slide26
or

= the moment generating function of the normal distribution with mean

and variance

Thus Y = a1X1 + … + anXnhas a normal distribution with mean a1m1+ …+ anmn and variance

slide27

Special case:

In this case X1, X2, …, Xn is a sample from a normal distribution with mean m, and standard deviations s, and

slide28

Thus

has a normal distribution with mean

and variance

slide29

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

slide31

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

slide32

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

slide33

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

slide38

Is the moment generating function of the standard normal distribution

Thus the limiting distribution of z is the standard normal distribution

Q.E.D.