Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

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

Uploaded on 08/04/2012

saroj
saroj 🇮🇳

4.5

(2)

169 documents

1 / 20

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
docsity.com
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14

Partial preview of the text

Download Probability and Stochastic Processes: Function of Two Random Variables and more Slides Probability and Stochastic Processes in PDF only on Docsity!

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

(z) of Z = min(X,Y)

Function of Two RV’s

Distance from origin to (x,y)

Z = (X

+ Y