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We have seen that the Bernoulli process can be expressed in several alternative ways, all
involving a sequence of rv’s. For each of these alternatives, there are generalizations that
lead to other interesting processes. A particularly interesting generalization arises by allowing the interarrival intervals to be arbitrary discrete or continuous nonnegative IID rv’s
14 This is one of those maddening arguments that, while essentially obvious, require some careful reasoning
to be precise. We go through those kinds of arguments with great care in the next chapter, and suggest that
skeptical readers wait until then. 16 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY rather than geometric rv’s. Letting the interarrival intervals be IID exponential rv’s leads
to the Poisson process, which is the topic of Chapter 2. IID Interarrival intervals with an
arbitrary distribution leads to renewal processes, which play a central role in the theory of
stochastic processes, and consitute the sub ject of Chapter 3.
Renewal processes are examples of discrete stochastic processes. The distinguishing characteristic of such processes is that interesting things (arrivals, departures, changes of state)
occur at discrete instants of time separated by deterministic or random intervals. Discrete
stochastic processes are to be distinguished from noise-like stochastic processes in which
changes are continuously occurring and the sample functions are continuously varying functions of time. The description of discrete stochastic processes above is not intended to be
precise. The various types of stochastic processes developed in subsequent chapters are all
discrete in the above sense, however, and we refer to these processes, somewhat loosely, as
discrete stochastic processes.
Discrete stochastic processes ﬁnd wide and diverse applications in operations research, communication, control, computer systems, management science, etc. Paradoxically, we shall
spend relatively little of our time discussing these particular applications, and rather spend
our time developing results and insights about these processes in general. We shall discuss a
number of examples drawn from the above ﬁelds, but the examples will be “toy” examples,
designed to enhance understanding of the material rather than to really understand the
application areas. 1.3.6 Expectations Before giving a general deﬁnition for the expected value (the expectation) of a rv, we discuss
several special cases. First, the expected value of a non-negative discrete rv X with PMF
pX is given by
E [X ] =
x pX (x).
x The expected value in this case must be ﬁnite if the rv has only a ﬁnite set of sample values,
but it can be either ﬁnite or inﬁnite if the rv has a countably inﬁnite set of sample values.
Example 1.3.2 illustrates a case in which it is inﬁnite. Next, assume X is discrete with both positiveP negative sample values. It is possible
in this case to have x≥0 x pX (x) = 1 and x<0 x pX (x) = −1, and in this case the
sum in (1.20) has no limiting value and the partial sums can range from −1 to +1
depending on the order in which the terms are summed. Thus the expectation is said to
be undeﬁned in this case. If one or both of these terms is ﬁnite, then the expectation is
given by (1.20). In this case, the expectation will be 1 if x≥0 x pX (x) = 1, will be −1
if x<0 x pX (x) = −1, and will be ﬁnite otherwise.
Many authors deﬁne expectation only in the case where x |x|pX (x) < 1, i.e., where both
the sum over negative sample values and positive sample values are ﬁnite. It is often useful,
however, to separate the case where both the positive and negative sums are inﬁnite from
the better behaved situations where at most one is inﬁnite. 1.3. PROBABILITY REVIEW 17 Example 1.3.2. This example will be useful frequently in illustrating rv’s that have an
inﬁnite expectation. Let N be a positive, integer-valued rv with the distribution function
FN (n) = n/(n + 1) for integer n ≥ 0. Then N is clearly a positive rv since FN (0) = 0 and
limN →1 FN (n) = 1. For each n ≥ 1, the PMF is given by
pN (n) = FN (n) − FN (n − 1) = n
n(n + 1) (1.21) P
Since pN (n) is a PMF, we see that 1 1/[n(n + 1)] = 1, which is a frequently useful fact.
The expected value for N , however, is inﬁnite since
E [N ] = 1
X n pN (n) = n=1 1
n(n + 1)
n=1 where we have used the fact that the harmonic series diverges.
A variation of this example is to let M be a random variable that takes on both positive
and negative values with the PMF
2|m| (|m| + 1) pM (m) = P
In other words, MP symmetric around 0 and |M | = N . Since n≥0 npN (n) = 1, it
follows easily that m≥0 mpM (m) = 1 and m<0 mpM (m) = −1. Thus the expectation
of M is undeﬁned.
For continuous rv’s, expected values are very similar to those for discrete rv’s. If a continuous
rv is positive with a probability density fX , the expected value can be either ﬁnite or inﬁnite
and is given by
E [X ] =
xfX (x) dx.
For an arbitrary continuous rv, the expectation...
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- Spring '09