Random Variable Transformation & Moments Calculation in Probability & Stochastic Processes, Slides of Probability and Stochastic Processes

Download Random Variable Transformation & Moments Calculation in Probability & Stochastic Processes and more Slides Probability and Stochastic Processes in PDF only on Docsity!

CS723 - Probability

and

Stochastic Processes

CS723 - Probability

and

Stochastic Processes

• Lecture No. 27Lecture No.

f

Y

(y) Directly from f

X

(x)

f

Y

(y) Directly from f

X

(x)

f

uv

(U,V) from f

Xy

(x,y)

f

Z

(z) from f

Xy

(x,y)

E[Y=g (X)] from f

X(

x)









 

 

 

 

















X 2 Y

Y 2 Y

2 Y

X

Y

Y

dx

(x) f

g(x)

dy

(y) f )

(y

dx

(x) f

g(x)

dy

(y) f y

E[Y=g(X)] from f

X

(x)

 

 

 

 

















X 2 Y

Y 2 Y

2 Y

X

Y

Y

dx

(x) f

g(x)

dy

(y) f )

(y

dx

(x) f

g(x)

dy

(y) f y

Some examples of simple transformations

2 X

2

2 Y

X

Y

2 X

2

2 Y

X

Y

2 X

2 Y

X

Y

x

y

x

y

x

y

Moments of U&V from f

XY

(x,y)

dy

dx

y)

(x,

f ) y , x ( g

y)

(x,

g

E

dy

dx

y)

(x,

f ) y x ( Y X E

XY XY

E[g(x,y)] using joint PDF f

XY

(x,y)

Moments of U&V from f

XY

(x,y)









dy

dx

y)

(x,

f ) y , x ( g

y)

(x, g

E

dy

dx

y)

(x,

f ) y x ( Y X E

XY XY



  







E[g(x,y)] using joint PDF f

XY

(x,y)













y)

(x, g

E

dy

dx

y)

(x,

f ) y , x ( g

y)

(x, g

E

2 2 U

(^2) U

XY

U

 











Moments of U&V from f

XY

(x,y)







dy

dx y)

(x,

f ) y , x ( h ) y , x ( g

)

V )(

U(

E

XY

V

U

V

U

UV



 

 



 

 



Similarly,  Examples of linear transformations functions

g(. ,. ) and h(. ,. )

x

u

y x v , y x u

To be continued …