Deriving Joint Probability Densities of Two Random Variables in Stochastic Processes, Slides of Probability and Stochastic Processes

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CS723 - Probability

and

Stochastic Processes

CS723 - Probability

and

Stochastic Processes

• Lecture No. 23Lecture No.

Multiple Derived RV’s

•^

U = g(X,Y) and V = h(X,Y) enable us to find

f^ U

(u) and f

(v) from fV

XY

(x,y)

•^

We need the joint density of U and V f

UV

(u,v)

to analyze joint behavior of U and V

-^

We can find f

UV

(u,v) from f

(u) and fU^

(v) if UV

and V are known to be independent

-^

If U and V are dependent, we have to find

f^ UV

(u,v) from f

XY

(x,y) , g(. , .) , and h(. , .)

•^

We can find f

(u) and fU^

(v) from fV

UV

(u,v) by

integrating along v-axis or u-axis

Pair of Derived RV’s

Red area represents event A where

A = {(X+Y) > 50

&^

|Y-X| < 6}

Probability Computation

.003 0

dx dy

e

e

(^005). 0

dx dy

e

e

(^005). 0

dx dy

e

dx dy

e

Pr(A)

28

6 x^6 - x

y/20-

x/10-

28 22

6 x

x- 50

y/20-

x/10-

28

6 x^6 - x

y)/ (2x-

28 22

6 x

x- 50

y)/ (2x-







  

  

  

  



  

  

  

  



 

^

^

^

^

 

^









Pair of Derived RV’s

Red area represents V > 50

|U| < 6

Computing F

UV

(u,v)

Computing F

UV

(u,v)

Surface Plot of F

UV

(u,v)

Surface Plot of f

UV

(u,v)