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Functions of Random Variables. Methods for determining the distribution of functions of Random Variables. Distribution function method Moment generating function method Transformation method. Distribution function method. Let X, Y, Z …. have joint density f ( x,y,z, … )

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Methods for determining the distribution of functions of random variables
Methods for determining the distribution of functions of Random Variables

  • Distribution function method

  • Moment generating function method

  • Transformation method


Distribution function method
Distribution function method Random Variables

Let X, Y, Z …. have joint density f(x,y,z, …)

Let W = h( X, Y, Z, …)

First step

Find the distribution function of W

G(w) = P[W ≤ w] = P[h( X, Y, Z, …)≤ w]

Second step

Find the density function of W

g(w) = G'(w).


Example student s t distribution
Example: Student’s t distribution Random Variables

Let Z and U be two independent random variables with:

  • Z having a Standard Normal distribution

    and

  • U having a c2 distribution with n degrees of freedom

Find the distribution of


The density of Random VariablesZ is:

The density of U is:


Therefore the joint density of Random VariablesZ and U is:

The distribution function of T is:


Then Random Variables

where


Student’s t distribution Random Variables

where


Student w w gosset
Student – W.W. Gosset Random Variables

Worked for a distillery

Not allowed to publish

Published under the pseudonym “Student


t distribution Random Variables

standard normal distribution



Let Random Variablesx1, x2, … , xndenote a sample of size n from the density f(x).

Let M = max(xi) then determine the distribution of M.

Repeat this computation for m = min(xi)

Assume that the density is the uniform density from 0 to q.


Hence Random Variables

and the distribution function






The probability integral transformation
The probability integral transformation Random Variables

This transformation allows one to convert observations that come from a uniform distribution from 0 to 1 to observations that come from an arbitrary distribution.

Let U denote an observation having a uniform distribution from 0 to 1.


Let Random Variablesf(x) denote an arbitrary density function and F(x) its corresponding cumulative distribution function.

Find the distribution of X.

Let

Hence.


Thus if Random VariablesU has a uniform distribution from 0 to 1. Then

has density f(x).

U



Definition
Definition Random Variables

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


The distribution of a random variable Random VariablesX 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 Random Variables

  • mX(0) = 1


  • Let Random VariablesX 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)


  • Let Random VariablesX 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





using Random Variables

or


then Random Variables



We will use distribution


Note: distribution

Also


Note: distribution

Also


Equating coefficients of distributiontk, we get


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 distribution of functions of Random Variables

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.


Thus distribution of functions of Random Variables 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 .


Example1
Example distribution of functions of Random Variables

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


or distribution of functions of Random Variables

= 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


Example2
Example distribution of functions of Random Variables

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


or distribution of functions of Random Variables

= 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


a distribution of functions of Random Variables = +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 distribution of functions of Random Variables (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


or distribution of functions of Random Variables

= 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


Special case: distribution of functions of Random Variables

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


Thus distribution of functions of Random Variables

has a normal distribution with mean

and variance


Summary distribution of functions of Random Variables

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


Sampling distribution of distribution of functions of Random Variables

Population


The Central Limit theorem distribution of functions of Random Variables

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


Proof: distribution of functions of Random Variables (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


Let distribution of functions of Random Variables 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


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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