Joint Probability Distributions & Correlation 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. 19Lecture No.

Exponential RV Pair

dy

dx

e

e

dy

dx

e

Pr(A)

20 0

y- 20 0

x/10-

y/20-

20 0

y- 20 0

y)/ (2x-

^



^

^

^



The joint PDF is f

XY

(x,y) = 0.005 e

-(2x+y)/

f^ XY

(x,y) = f

(x) fX

(y) = (0.1eY

-x/

)(0.05e

-y/

A = {(x,y) s.t. x+y < 20}B = {(x,y) s.t. x+y > 20 & x > y}

^



^



 

^



^





^









dy dx e

e

dy dx e

e

(^005). 0

dy dx

e

dy dx

e

Pr(B)

10

y

x/10-

y/20-

10 0

y- 20

x/10-

y/20-

10

y

y)/ (2x-

10 0

y- 20

y)/ (2x-

Exponential RV Pair

Pair of Dependent RV’s

f^ XY

(x,y) defined for (x,y)

ε^

[0,10]x[0,10]

f^ XY

(x,y) = K(2x + y) = (2x + y)/

f^ (x) = (5 + 2x)/150X

&^

f^ (y) = (10 + y)/150Y

Trajectories on Joint PDF 0

1

2

3

4

5

6

7

8

9

10

Trajectories on Joint PDF 0

1

2

3

4

5

6

7

8

9

10

Pair of Dependent RV’s

fXY

(x,y) defined for (x,y)

ε^

[0,

∞)x[0,

∞

)

fXY

(x,y) = K( e

-x/

e

-y/

e

-xy/

)

Portion of trajectories are shown

0

2

4

6

8

10

12

(^1) 0.8 0.6 0.4 0.2 0

Trajectories on Joint PDF

-^15 -^ -^ -^ -^

0

3

6

9

12

15

Correlation & Covariance • Joint PDF hold the complete informationabout the behavior of two RV’s • The means and variances are importantcharacteristics of individual RV’s • Correlation is a mixed higher moment thatdescribes relationship between two RV’s • Positive (negative) correlation implies that ifone random variable assumes a large value,then the other is more likely to assume alarge (small) value • No correlation between independent RV’s