Chapter 4 Jointly distributed Random variables

RS ? 4 ? Jointly Distributed RV (a)

Chapter 4 Jointly distributed Random variables

Continuous Multivariate distributions

(for private use, not to be posted/shared online)

Continuous Random Variables

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RS ? 4 ? Jointly Distributed RV (a)

Joint Probability Density Function (pdf)

Definition: Joint Probability density function Two random variable are said to have joint probability density function , if

1. , 0.

2.

, 1.

3. ,

,

Marginal and Condition Density

Definition: Marginal Density Let and denote two RVs with joint pdf , , then the marginal density of is

,

and the marginal density of is

,

Definition: Conditional Density Let and denote two RVs with joint pdf , and marginal densities , , then the conditional density of given = and the conditional density of given = are given by

fY X y

x

f x, y fX x

f X Y x

y

f x, y fY y

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RS ? 4 ? Jointly Distributed RV (a)

Joint MGF

Definition: MGF of (X,Y) Let X and Y be two RVs with joint pdf , , then the MGF of X Y:

mXY (t1, t2 ) E[exp(t1 X t2Y )] exp(t1 X t2Y ) f (x, y)dxdy R2

Theorem: The MGF of a pair of independent RVs is the product of the MGF of the corresponding marginal distributions. That is,

, =

Proof: m XY (t1 , t2 ) exp( t1 X t2Y ) f ( x, y )dxdy

exp( t1 X ) exp( t2Y ) f ( x) f ( y )dxdy

exp( t1 X ) f ( x)dx exp( t2Y ) f ( y )dy m X (t1 )mY (t2 )

Marginal MGF

Definition: MGF of the marginal distribution of X (andY) Let , be the MGF of (X,Y), then the MGF of the marginal distributions of X and Y are, respectively, , 0 and 0,

Proof:

m X (t) exp(tX ) f X (x)dx exp(tX )[ f XY (x, y)dy]dx

exp(tX ) f XY (x, y)dydx mXY (t,0)

Similar derivation for Y.

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RS ? 4 ? Jointly Distributed RV (a)

The bivariate Normal distribution

Sir Francis Galton (1822 ?1911, England)

The bivariate normal distribution

Let the joint distribution be given by:

f

x1 , x2

2

1

1 2

e

1 2

Q

x1

,

x2

1 2

where

Q

x1 , x2

x1 1 1

2

2

x1 1 1

x2 2 2

1 2

x2 2 2

2

This distribution is called the bivariate Normal distribution.

The parameters are 1, 2 , 1, 2 and The properties of this distribution were studied by Francis Galton and discovered its relation to the regression, term Galton coined.

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RS ? 4 ? Jointly Distributed RV (a)

The bivariate normal distribution

Surface Plots of the bivariate Normal distribution

The bivariate normal distribution

Note: We can have a more compact joint using linear algebra:

f

(x1,

x2

)

21

2

1

(1

2

)

exp

1 2(1

2

)

(

x1 1 1

)2

2(

x1 1 1

)(

x2 2 2

)

(

x2 2 2

)2

2

2

1 (|

|)1/

2

exp

1 2

(x

)'1(x

)

(1) Determine the inverse and determinant of (the covariance matrix)

1122

2221

|

|

12

2 2

122

12

2 2

(1

2 12

12

2 2

)

12

2 2

(1

2)

1

|

1

|

2212

1221

5

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