THE GEORGE WASHINGTON UNIVERSITY
SCHOOL OF ENGINEERING AND APPLIED
SCIENCE
ECE 6015 STOCHASTIC PROCESSES IN
ENGINEERING
LECTURE NOTES
HERMANN J. HELGERT
PROFESSOR OF ENGINEERING AND APPLIED
SCIENCE
ECE 6015 Stochastic Processes in Engineering
1
PART A
RAN
SECTION A-4
BOUNDS ON RANDOM VARIABLES
ECE 6015 Stochastic Processes in Engineering
1
THE MARKOV INEQUALITY
GIVEN THE RANDOM VARIABLE X WITH SX = cfw_ 0 x
fX (x) = 0 FOR x ; IS A CONSTANT
THEN:
E(X) = = +
= P(x )
P(x ) E(X)/
ECE 6015 Stochastic Processes
SECTION A-3
MULTIPLE RANDOM VARIABLES
ECE 6015 Stochastic Processes in Engineering
1
TWO DISCRETE RANDOM VARIABLES
X AND Y ARE TWO DISCRETE RANDOM VARIABLES WITH VALUE SETS
S X = cfw_ x 1 , x 2 , x 3 , x m ; S Y = cfw_ y 1 , y2 , y 3 , y n
THE JOINT PRO
SECTION A-2
RANDOM VARIABLES
ECE 6015 Stochastic Processes in Engineering
1
RANDOM VARIABLES
GIVEN AN EXPERIMENT WITH A SAMPLE SPACE OF POSSIBLE OUTCOMES
cfw_S AND A SET OF EVENTS:
A RANDOM VARIABLE X IS AN ASSIGNMENT OF A REAL NUMBER x(A) TO
EACH OF THE
THE GEORGE WASHINGTON UNIVERSITY
SCHOOL OF ENGINEERING AND APPLIED
SCIENCE
ECE 6015 STOCHASTIC PROCESSES IN
ENGINEERING
LECTURE NOTES
HERMANN J. HELGERT
PROFESSOR OF ENGINEERING AND APPLIED
SCIENCE
ECE 6015 Stochastic Processes in Engineering
1
PART B
RAN
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #3 Solutions
Problem 1
4.12)
(b) P[x<=-1]=.5
P[x=-1]=.5
P[x<.5]=P[x<=.5]-p[x=.5]=
- 0= .75
P[-.5<x<.5]=P[x<=.5]-P[x=.5]-P[x<=-
School of Engineering and Applied Science
The George Washington University
ECE 6015
Stochastic Processes in Engineering
Homework #4 Solutions
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5