Week 91 Example A device containing two key components fails when and only when both components fail. The lifetime, T 1 and T 2, of these components are.

week 9 1

Example

• A device containing two key components fails when and only when both components fail. The lifetime, T1 and T2, of these components are independent with a common density function given by

• The cost, X, of operating the device until failure is 2T1 + T2. Find the density function of X.

otherwise

tetf

t

T0

0

week 9 2

Convolution

• Suppose X, Y jointly distributed random variables. We want to find the probability / density function of Z=X+Y.

• Discrete case

X, Y have joint probability function pX,Y(x,y). Z = z whenever X = x and Y = z – x. So the probability that Z = z is the sum over all x of these jointprobabilities. That is

• If X, Y independent then

This is known as the convolution of pX(x) and pY(y).

x

YXZ xzxpzp .,,

x

YXZ xzpxpzp .

week 9 3

Example

• Suppose X~ Poisson(λ1) independent of Y~ Poisson(λ2). Find the

distribution of X+Y.

4

Convolution - Continuous case• Suppose X, Y random variables with joint density function fX,Y(x,y). We want to

find the density function of Z=X+Y.Can find distribution function of Z and differentiate. How?The Cdf of Z can be found as follows:

If is continuous at z then the density function of Z is given by

• If X, Y independent then

This is known as the convolution of fX(x) and fY(y).

z

v x

YX

x

z

v

YX

x

xz

y

YXZ

dxdvxvxf

dvdxxvxf

dydxyxfzYXPzF

.,

,

,

,

,

,

x

XY dxxvxf ,

x

XYZ dxxzxfzf ,

x

YXZ dxxzfxfzf

week 9 5

Example

• X, Y independent each having Exponential distribution with mean 1/λ. Find the density for W=X+Y.

week 9 6

Some Recalls on Normal Distribution

• If Z ~ N(0,1) the density of Z is

• If X = σZ + μ then X ~ N(μ, σ2) and the density of X is

• If X ~ N(μ, σ2) then

zezz

Z ,2

1 2

2

xexf

x

X ,2

1 2

2

2

2

.1,0~ NX

Z

week 9 7

More on Normal Distribution

• If X, Y independent standard normal random variables, find the density of W=X+Y.

week 9 8

In general,

• If X1, X2,…, Xn i.i.d N(0,1) then X1+ X2+…+ Xn ~ N(0,n).

• If , ,…, then

• If X1, X2,…, Xn i.i.d N(μ, σ2) then

Sn = X1+ X2+…+ Xn ~ N(nμ, nσ2) and

2111 ,~ NX 2

222 ,~ NX 2,~ nnn NX

.,~ 221121 nnn NXXX

.,~2

nN

n

SX n

n

9

Sum of Independent χ2(1) random variables

• Recall: The Chi-Square density with 1 degree of freedom is the

Gamma(½ , ½) density.

• If X1, X2 i.i.d with distribution χ2(1). Find the density of Y = X1+ X2.

• In general, if X1, X2,…, Xn ~ χ2(1) independent then

X1+ X2+…+ Xn ~ χ2(n) = Gamma(n/2, ½).

• Recall: The Chi-Square density with parameter n is

otherwise

xxenxf

nx

n

X

0

0

2

2

11

22

1

2/

week 9 10

Cauchy Distribution

• The standard Cauchy distribution can be expressed as the ration of two Standard Normal random variables.

• Suppose X, Y are independent Standard Normal random variables.

Let . Want to find the density of Z. X

YZ

week 9 11

Change-of-Variables for Double Integrals• Consider the transformation , u = f(x,y), v = g(x,y) and suppose we are

interested in evaluating .

• Why change variables?In calculus: - to simplify the integrand.

- to simplify the region of integration.In probability, want the density of a new random variable which is a function of other random variables.

• Example: Suppose we are interested in finding .

Further, suppose T is a transformation with T(x,y) = (f(x,y),g(x,y)) = (u,v). Then,

• Question: how to get fU,V(u,v) from fX,Y(x,y) ?

• In order to derive the change-of-variable formula for double integral, we need the formula which describe how areas are related under the transformation T: R2 R2 defined by u = f(x,y), v = g(x,y).

Dxy

xydAyxF ,

.,,A

YX dxdyyxfAP

.,,AT

VU dudvvufAP

week 9 12

Jacobian

• Definition: The Jacobian Matrix of the transformation T is given by

• The Jacobian of a transformation T is the determinant of the Jacobian matrix.

• In words: the Jacobian of a transformation T describes the extent to which T increases or decreases area.

yx

vu

y

g

x

gy

f

x

f

yxJT ,

,,

week 9 13

Change-of-Variable Theorem in 2-dimentions

• Let x = f(u,v) and y = g(u,v) be a 1-1 mapping of the region Auv onto Axy with f, g having continuous partials derivatives and det(J(u,v)) ≠ 0 on Auv. If F(x,y) is continuous on Axy then

where

AuvAxy

dudvvuJvugvufFdxdyyxF ,,,,,

yxJv

y

u

yv

x

u

x

vuJ,

1,

week 9 14

Example

• Evaluate where Axy is bounded by y = x, y = ex, xy = 2 and xy = 3.Axy

xydxdy

week 9 15

Change-of-Variable for Joint Distributions• Theorem

Let X and Y be jointly continuous random variables with joint density

function fX,Y(x,y) and let DXY = {(x,y): fX,Y(x,y) >0}. If the mapping T given

by T(x,y) = (u(x,y),v(x,y)) maps DXY onto DUV. Then U, V are jointlycontinuous random variable with joint density function given by

where J(u,v) is the Jacobian of T-1 given by

assuming derivatives exists and are continuous at all points in DUV .

otherwise

DvuifvuJvuyvuxfvuf VUYX

VU0

,,,,,, ,,

,

v

y

u

yv

x

u

x

vuJ

,

week 9 16

Example• Let X, Y have joint density function given by

Find the density function of

otherwise

yxifeyxf

yx

YX0

0,,,

.YX

XU

week 9 17

Example

• Show that the integral over the Standard Normal distribution is 1.

week 9 18

Density of Quotient

• Suppose X, Y are independent continuous random variables and we are interested in the density of

• Can define the following transformation .

• The inverse transformation is x = w, y = wz. The Jacobian of the inverse transformation is given by

• Apply 2-D change-of-variable theorem for densities to get

• The density for Z is then given by

.X

YZ

xwx

yz ,

wwz

z

y

w

yz

x

w

x

zwJ

01

,

wwzfwfwwzwfzwf YXYXZW ,, ,,

dwwwzfwfzf YXZ

week 9 19

Example

• Suppose X, Y are independent N(0,1). The density of is X

YZ

week 9 20

Example – F distribution

• Suppose X ~ χ2(n) independent of Y ~ χ2

(m). Find the density of

• This is the Density for a random variable with an F-distribution with parameters n and m (often called degrees of freedom). Z ~ F(n,m).

./

/

mY

nXZ

week 9 21

Example – t distribution

• Suppose Z ~ N(0,1) independent of X ~ χ2(n). Find the density of

• This is the Density for a random variable with a t-distribution with parameter n (often called degrees of freedom). T ~ t(n)

.

nX

ZT

week 9 22

Some Recalls on Beta Distribution

• If X has Beta(α,β) distribution where α > 0 and β > 0 are positive parameters the density function of X is

• If α = β = 1, then X ~ Uniform(0,1).

• If α = β = ½ , then the density of X is

• Depending on the values of α and β, density can look like:

• If X ~ Beta(α,β) then and

otherwise

xxxxf X

0

101 11

otherwise

xforxxxf X

0

101

1

XE

.12

XV

week 9 23

Derivation of Beta Distribution

• Let X1, X2 be independent χ2(1) random variables. We want the density of

• Can define the following transformation

21

1

XX

X

21221

11 , XXY

XX

XY