CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
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Probability and Stochastic Processes: Function of Two Random Variables, Slides of Probability and Stochastic Processes
Lecture notes for cs723 on probability and stochastic processes, focusing on the concept of a function of two random variables. It covers the evaluation of the cumulative distribution function (cdf) and probability density function (pdf) for various functions such as the sum, difference, ratio, product, maximum, and minimum of two random variables. Examples and events for each function are provided.
Typology: Slides
2011/2012
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CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
- Lecture No. 22Lecture No.
Function of Two RV’s
Ratio of service times of two TV’s Z = Y / X
Evaluation of FZ(z)
0
1
2
3
4
5
6
7
8
9
10
0 0.5 0.4 0.3 0.2 0.
Probability Density Function f
Z
(z) of Z = Y / X
PDF of Z = Y / X
f
Z
(z) = 1/(2+z) – z/(2+z)
Events for Z = Y / X
Pr( 0.5 < Z )
Pr( Z > 2 )
Evaluation of FZ(z)
Evaluation of FZ(z)
The double integral to get F
Z
(z) is given by
dx
dy
e
e
005 .
0
dx
dy
e
(z)
F
0
z/x
0
y/
x/
0
z/x
0
y)/
(2x
Z
Evaluation of FZ(z)
The parameter z only takes on positive valuesThe area of integration is always a squareDue to independence, the probability of this squareis easy to find
3z/
z/
z/
z/
z/
Y
X
Z
e - 1 e - 1
(z)
F
(z)
F
(z)
F
CDF of Z = max(X,Y)
-3z/
-z/
-z/
Z
e e - e - 1
(z)
F
0
10
20
30
40
50
60
70
80
90
100
0 0.8 0.6 0.4 0.
1
Cumulative Distribution Function F
Z
(z) of Z = max(X,Y)
Events for Z = max(X,Y)
Pr( Z < 3 )
Pr( Z > 30)
Function of Two RV’s
Min of service times of two TV’s Z = min(X,Y)
CDF of Z = min(X,Y)
-3z/
Z
e
1
(z)
F
0
10
20
30
40
0 0.8 0.6 0.4 0.
1
Cumulative Distribution Function F
Z