Two Random Variables - Stochastic Structural Dynamics - Lecture Slides, Slides of Stochastic Processes

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Multi-dimensional random variables-

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Recall •Two random variables

•Joint PDF•Joint pdf•Conditional PDF•Conditional pdf & Conditional expectations•Independence of RVs•Joint Expectations •Correlation function•Functions of random variables

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4

^

^

^



r

r r

J

r

y

r

x

r

y x y x y x

p^ XY

^  









  



 ^ cos

sin

sin

cossin cos

exp

1

2 2

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^

^

^

^













 

2

(^0) ;

0 ; 2

exp 2

sin

cos

1 2

exp 2

| |

,

,

2

2 2 2

2 cossin

    

  

  



 

^ 





 

 



r

r

r

r

r

r

J

y x p

r p

r y

r x

XY

R Joint pdf

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Typical Rayleigh pdf Typical uniform pdf

  rp R

r

  p^



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^

^

^



^

^

^



^

. ,

Deptermine

2 sin

ln 2

2 cos

ln 2

1.Define to 0 in

variables

random

d

distribute

uniformly and t,

independen be

and

Let

tion)

Transforma

Muller-

(Box

(^1212)

v u p

Y

X

V

Y

X

U

Y

X

UV

 

  Example

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^

) (^1) , (^0) ( ~ &) (^1) , 0 ( ~ &

,

; 2

exp (^12) ,

2

exp (^12)

1

1

1 (^12)

1

1 (^12)

) ( 2

exp

2

exp

2 2

2 2

2

2 2

2 2

2 2

1

N V N U V U

v u v u v u p

v u

u v u

v u

v u

v v u

v u

J UV

 



   

^  

 



  

^  

 

           

          

  

^  

 

  

^  

 

 







 Box-Muller Transformation (continued)

This transformation could be used in simulation of Gaussian random numberson a computer (more on this later).

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^



t.

independen are

and

if

check and ,

Determine

1 0

0 1

0 0

&

ExampleGiven

2

2

V

U

v u p

~N X Y

X Y V

Y

X U

UV

  

  

  

        

     







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^

^

^



 

^



 

^

^

^



^

^

 

 

d.

distribute

Cauchy is

and d

distribute ly

exponetial is

exp (^12)

with

exp

(^112)

2

0

2

2

2

2

2

V

U

V

U

v p u p v u p

v

v

du v u p

v p

u

u

dv v u p

u p

v

u

u

v

v u p

V

U

UV

UV

V

UV

U UV

 

   











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^



 

able.

dummy varia

Introduce

.

variables

random two

of

function one with

dealing are we Here:

Note

.

Determine

,

~

&

ExampleGiven Strategy

u p

y x p

X Y

Y X U

U

XY      

 

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 

^

^

^

^

^



 

 

^

^

 

dv v p v u p u p

Y V u p Y X U

x,y

by

b y p

ax

a x p

Y

X

U

U

Y

X

  

able

dummy vari the

Introduce

Find.

Define

with

exp

exp

Example Let Solution

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 

^

^

 

^

^

^



^



 

^

^

^



^

^

^



^



^

^

^



^

^











 

u

bv

au

a ab b

dv v u a

a bv

b

dv v u p bv

b

u p

v u

v u p

dv v u p bv

b

dvv p v u p u p

u

u

X

U

X

X

Y

X

U

exp

exp

exp

exp

exp

Since

exp 0

0

0

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^

^

^

^

^

^

^

^

1 1 2

1

1 ,^

,^ ,

Let

be a set of random variables.

where

.

n

n i i n^

n

i^

i^

i

X^

i

i n X^

X^

X

X X^

X

n^

X

P^

x^

P^

X^

x^

x^

x

n^

X

P^

x

p^

x





 

^



^

^



^



^



^



 

More than one random variables^  - th order Joint pdf of









^

 





De

f i n

i t i o n

s

• t

h o

r d

e r

J o

i n t

PD

Fo

f

^

^



^

^

^

^

^

2

1

2

,^

,^

,^ n n

X

x^

x

x^

x^

x

g^

X

g^

X^

g^

x p

x dx n

 

^

^



^



Expectation ofNote : The above is an

- fold integral. 

  

^

^

^



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^

^

1

2

1

2

Let

be a vector of random variables.

Let

such that the mean vector is given by

.

The elements of covariance matrix

is given by

t n

i^

i

t n

ij^

i X^

X^

X^

X

m^

X

m^

m^

m^

m

C

C^

  X   Multi - dimensional Gaussian random variable

 

^

^



^

^

^

^

^

^

^

^

1

(^12) 1

2

1

2

1

2

1

2

2

; clearly

and hence

.

Assume that

exists.

Let

be a realization of

.

is said to be Gaussian distributed if

1 2

n

t

i^

j^

j^

ij^

ji

-

t^

t

n^

n

t n

X^

X ,X

,^

,X^

n

n

m^

X^

m^

C^

C^

C^

C

C

x^

x^

x^

x^

X^

X^

X^

X

X^

X^

X^

X

p^

x^

p^

x ,x ,

,x

π

^

^

^



^







Definition 

 





 ^

^

^

^



1

1 2

1 exp

1 2

2

t

i

x^

m^

C^

x^

m^

;^

x^

i^

, ,

,n

C



^



^

^

^

 

  

^



^

^



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