Functions of random variables
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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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Functions of Random Variables

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Functions of random variables

Functions of Random Variables


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

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

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


Functions of random variables

The density of Z is:

The density of U is:


Functions of random variables

Therefore the joint density of Z and U is:

The distribution function of T is:


Functions of random variables

Then

where


Functions of random variables

Student’s t distribution

where


Student w w gosset

Student – W.W. Gosset

Worked for a distillery

Not allowed to publish

Published under the pseudonym “Student


Functions of random variables

t distribution

standard normal distribution


Distribution of the max and min statistics

Distribution of the Max and Min Statistics


Functions of random variables

Let x1, 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.


Functions of random variables

Hence

and the distribution function


Functions of random variables

Finding the distribution function of M.


Functions of random variables

Differentiating we find the density function of M.

f(x)

g(t)


Functions of random variables

Finding the distribution function of m.


Functions of random variables

Differentiating we find the density function of m.

f(x)

g(t)


The probability integral transformation

The probability integral transformation

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.


Functions of random variables

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

Find the distribution of X.

Let

Hence.


Functions of random variables

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

has density f(x).

U


Use of moment generating functions

Use of moment generating functions


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


Functions of random variables

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


Functions of random variables

  • 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)


Functions of random variables

  • 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


M g f s continuous distributions

M. G. F.’s - Continuous distributions


M g f s discrete distributions

M. G. F.’s - Discrete distributions


Moment generating function of the gamma distribution

Moment generating function of the gamma distribution

where


Functions of random variables

using

or


Functions of random variables

then


Moment generating function of the standard normal distribution

Moment generating function of the Standard Normal distribution

where

thus


Functions of random variables

We will use


Functions of random variables

Note:

Also


Functions of random variables

Note:

Also


Functions of random variables

Equating coefficients of tk, 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

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.


Functions of random variables

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 .


Example1

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


Functions of random variables

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


Example2

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


Functions of random variables

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


Functions of random variables

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


Functions of random variables

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


Functions of random variables

Special case:

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


Functions of random variables

Thus

has a normal distribution with mean

and variance


Functions of random variables

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


Functions of random variables

Sampling distribution of

Population


Functions of random variables

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


Functions of random variables

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


Functions of random variables

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


Functions of random variables

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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