Joint Probability Density and Dependence of Derived Random Variables U and V, Slides of Probability and Stochastic Processes

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CS723 - Probability

and

Stochastic Processes

CS723 - Probability

and

Stochastic Processes

• Lecture No. 24Lecture No.

Event A on f

UV

(u,v)

Probability Computation

dv

e

dv

du

e

e

dv

du

e

Pr(A)

50

3v/

50

(^66) -

u/

3v/

50

(^66) -

)/ v 3

u (







 







Independence of U and V





10 / v

20 / v

v

v

)/ v 3 - u ( V

e

e

1 .

0

du

e

(v)

f















u (

e

dv

e

u (

e

dv

e

(u)

f

20 / u

u

)/ v 3

u (

10 / u

u

)/ v 3 - u ( U



    





) v , u ( f e e 1

(v)

f

(u)

f

UV

10 / ) v

u (

20 / ) v u 2 ( V U





Pair of Derived RV’s

Red area represents event B where

B = {(X+Y) > 50

&

0.5 < (Y/X) < 2}

Probability Computation

dx

dy

e

e

005 .

0

dx

dy

e

e

005 .

0

dx

dy

e

dx

dy

e

Pr(B)

100/

2x x/

y/

x/

100/ 50/

2x

x

50

y/

x/

100/

2x x/

y)/

(2x

100/ 50/

x 2

x

50

y)/

(2x

  

  

  

  



  

  

  

  































Computing F

UV

(u,v)

Surface Plot of F

UV

(u,v)

Surface Plot of f

UV

(u,v)

Event B on f

UV

(u,v)

Probability Computation

dv

du

) 1 u ( e v

dv

du

e

u

v

Pr(B)

50

2

2

) 1

u (

20

) 2

u ( v

50

2

) 1

u (

20

) 2 u ( v 2 





  





 









 

