CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
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Function of Two Random Variable-Probability and Stochastic Processes-Lecture Slides, Slides of Probability and Stochastic Processes
Probability and Stochastic Processes course is part of basic science because of its usage in many fields. Most of its concepts are explained by using common examples like coin toss, rolling dice, deck of cards. Prof Mayur Somnath delivered this lecture to discuss Behaviour, Density, Marginal, Correlation, Covariance, Correlation, Coefficient, Function, Random
Typology: Slides
2011/2012
1 / 18
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CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
- Lecture No. 21Lecture No.
Function of Two RV’s
-^
A single value obtained from observed values of tworandom variables
-^
The two observed values are combined through afunction of two variablesThe value of the function is another randomvariable defined on the same sample space
-^
Combined service time of two televisionsSafe travel on a jet plane with two enginesThe distance from origin of random point (x,y)Examples: Z = X+Y , Z = X * Y ,
Z = max(X,Y)
Z = X – Y , Z = X/Y ,
Z = min(X,Y)
Function of Two RV’s
Combined service time of two TV’s Z = X + Y
Evaluation of F
(z)Z^
The double integral to get F
(z) is given byZ
z/10-
z/20- z 0
y- z^0
x/10-
y/20-
z 0
y- z^0
y)/
(2x-
Z
dy
dx
e
e
dy
dx
e
(z)
F
^
CDF of Z = X + Y
F
(z) = 1 – 2eZ
-z/
- e
-z/
z <
0
10
20
30
40
50
60
70
80
90
100
110
120
(^1) 0.8 0.6 0.4 0.2 0
Cumulative Distribution Function f
(z) of Z = X + YZ
Events for Z = X + Y
Pr(24 < Z
Pr( Z > 60)
Function of Two RV’s
Difference of service times of two TV’s Z=Y - X
Evaluation of F
(z)Z^
For negative values of z, the double integral to getF
(z) is given byZ
(^10) / z
z-
x z^0
y/20-
x/10-
z-
x z^0
y)/ (2x-
Z
e
dx
dy
e
e
dx
dy
e
(z)
F
^
^
^
Evaluation of F
(z)Z^
If z > 0, thevalue of F
(z) isZ
the volume ofthe sand hillbounded by +x-axis, +y-axisand line y = z+x
CDF of Z = Y - X
F
(z) = (1/3) eZ
z/
< z < 0
F
(z) = 1 – (2/3) eZ
-z/
≤^
z <
0
10
20
30
40
50
60
70
80
90
100
(^1) 0.8 0.6 0.4 0.2 0
Cumulative Distribution Function f
(z) of Z = Y - XZ
PDF of Z = Y - X
f^ Z
(z) = (1/30) e
z/
< z < 0
f^ Z
(z) = (1/30) e
-z/
z <
0
10
20
30
40
50
60
70
80
90
100
0 0.0350.03 0.0250.02 0.0150.01 0.
Probability Density Function f
(z) of Z = Y - XZ