Chapter six
Random Processes
Random Processes
Chapter six
Random Processes
Chapter six
Random Processes
SPECIFYING A RANDOM PROCESS
Joint Distributions of Time Samples
as shown in Fig. 9.1. The joint behavior of the random process at these k
time instants
Chapter 9
Markov chains
THE ELEMENTS OF A QUEUEING SYSTEM
Figure 12.1(a) shows a typical queueing system and Fig. 12.1(b) shows the
elements of a queueing system model.
Customers from some population arrive at the system at the random arrival times
S1 , S
Chapter 4 part 2
FUNCTIONS OF A RANDOM VARIABLE
1
Chapter 4 part 2
2
Chapter 4 part 2
3
Chapter 4 part 2
4
Chapter 4 part 2
5
Chapter 4 part 2
the Chebyshev
inequality can give
rather loose bounds.
Nevertheless, the
inequality is useful in
situations in w
Chapter 4 part 2
Conditional Expectation
The conditional expectation of Y given X=x is defined by
In the special case where X and Y are both discrete random variables we
have:
Chapter 4 part 2
Multiple Random Variables
Chapter 4 part 2
Joint Distribution
Chapter 8
MARKOV PROCESSES
11.1 MARKOV PROCESSES
A random process X(t) is a Markov process if the future of the
process given the present is independent of the past, that is, if for
arbitrary times t1 < t2 < < tk < tk+1 ,
if X(t) is discrete-valued,
if X(
CHAPTER 5
Random Variables and Long-Term Averages
Sums of
SUMS OF RANDOM VARIABLES
Let X1, X2, Xn be a sequence of random variables, and let Sn
be their sum: Sn = X1 + X2 + + Xn
(5.1)
The Mean Value: the expected value of a sum of n random variables is eq
Chapter 6 2nd Part.
Random Processes
CONTINOUS-TIME RANDOM PROCESSES
Poisson Process
Consider a situation in which events occur at random instants of time at
an average rate of events per second.
For example, an event could represent the arrival of a cust
Chapter 4
Random Variable
One
THE CUMULATIVE DISTRIBUTION FUNCTION
The probability mass function of a discrete random variable was defined
in terms of events of the form cfw_X = b.
The cumulative distribution function is an alternative approach which
uses
Chapter 3
Discrete Random Variables
A random variable X is a function that assigns a real number
X(), to each outcome in the sample space of a random
experiment. Recall that a function is simply a rule for assigning a
numerical value to each element of a
Basic Concepts of Probability Theory
SPECIFYING RANDOM EXPERIMENTS
A random experiment is an experiment in which the outcome varies in
an unpredictable fashion when the experiment is repeated under the
same conditions. A random experiment is specified by