This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1
FUNDAMENTALS OF PROBABILITY The foundations of the science of statistics is mostly based on probability theory.
Probability is also often used in decisionmaking applications in which risk or incomplete
information is involved. The beginnings of probability theory were originated in games of chance, say, tossing
coins, dealing cards, and rolling dice. From such beginnings, an extensive mathematical theory
has evolved. R. A. Fisher and R. Von Mises were largely responsible for initiating the theory
[Mises]. The modern axiomatic treatment of probability is due to A. N. Kolmogorov [Kolmogorov]. This chapter discusses the fundamentals of probability theory. We will learn that
probability is a mathematical model to understand physical systems in an average sense. The
concepts of probability will be introduced by considering discrete random events. In the
succeeding chapters, this will be extended to continuous random variables, random functions of tim e, and to the analysis of the response of linear systems to random inputs. PROBABILITY AND RANDOMNESS
Probability deals with averages of phenomena which involve random behavior.
Randomness is uncertainty in some form. For example, when tossing dice, drawing cards and
spinning roulette wheels, one is uncertain of the outcome in advance.
In such random phenomena and random systems, certain averages approach a constant
value as the number of observations increases. For example, in tossing the coin, the percentage of heads approaches 0.5 as the coin is tossed many times. Probability is a mathematical model to investigate random physical systems in an average
sense. Most physical systems involve a certain degree of randomness. For example, Ohm‘s
law in electrical engineering is given by 120‘) = Rf“) (1.1)
where v(!) is the instantaneous voltage, :‘(r) is the instantaneous current, and R is the resistance
of the medium. Eq. (1.1) is true only in an average sense. In reality, both z'(t) and 12(1) have
random components. Similarly, many famous laws of physics are true only in an average sense. A good understanding of randomness is necessary before embarking on studying the
abstract theories of probability. Below we discuss three signals for this purpose. The ﬁrst
signal is given by x(r)=10cosd2nt+45°) (1.2)
and is shown in Figure 1. This signal is a sinusoid with a frequency of 60 Hz and a phase of 45". It is perfectly deterministic, and hence has no randomness about it. Signal The second signal is generated by a “noise generator” rePresenl'ed by a random number
generator in a digital computer, and is showu in Figure 2. It is seen that it has irregular
behavior, unlike Figure 1. It appears impossible to determine the values of the signal outside the region shown in the ﬁgure. We conclude that this signal should be studied probabilistioally. Signal However, the randomness of a signal can be an illusion due to lack of knowledge on the
signal. This is especially true of signals governed by deterministic chaos. Chaotic signals look
like random signals, but they are actually governed by deterministic laws. Chaos can be
deﬁned as aperiodic longterm behavior in a deterministic system which exhibits sensitive
dependence on initial conditions. For example, consider the logistic map generated by xn+1 ' 3’91“ Xn) (13)
where x” 2 0. Eq. (1.3) is the discretetime analog of the logistic equation for population
growth. Suppose that a is ﬁxed, and x” is generated ﬁ‘om Eq. (1.3) iteratively, starting with an initial value x0. For a 4 1, x” goes towards O as a inereases. For 1 é a < 3, the population grows and eventually reaches a steadystate. An example is showu in Figure 3 for a = 2.9 and x(0) = 0.77. However, as a approaches 4, xle as a ﬁlncl'ion ofn looks more and more chaotic. An example is shown in Figure 4 with a = 4.0 and x(0) = 0.77. The signal of Figure 1.4 is a chaotic signal, but not a random signal. Signal 0.8 0.75 0.7 0.6 0.55 i 0.5 0.1 _l l _J .1 I—
0 3 0.4 0.5 0 6 0.7 0.8 0.9
Time 0.9 0.8 0.2 0.3 0.4 0.5
Time Lima. 0.6 0.7 0.8 0.9 EVENTS AND RANDOM EXPERIMENTS
A random experiment is an experiment whose outcome (end result) is not know in
advance. For example, ﬂipping a coin is a random experiment. The possible outcomes are
heads or tails, which are not known in advance. An event is one or a number of the possible outcomes of the random experiment. The concept of equally likely events is important in probability. For example, in selecting
a card from a deck of cards, the event of selecting a particular card is equally likely among 52
candidate events. A term often used synonomously with the concept of equally likely events is
the concept of selecting at random. For example, if a card is selected at random from a deck of cards, all cards in the deck are equally likely to be selected. A trial is a single performance of an experiment. An elementary event is an event for
which there is only one possible outcome. For example, in tossing the coin, the only possible
outcomes are heads (H) or tails (T) which can not both occur. Similarly, when a dice is rolled, the event of getting one of the integers from 1 to 6 is an elementary event. An event can be deﬁned in such a way that it involves more than one outcome. Then, it is
a composite event. For example, the event of getting an even integer when rolling a dice is a
composite event since it involves the outcomes 2, 4 or 6. An impossibie event is an event whose probability (of occurrence) is zero. A certain event is an event whose probability of occurrence is one. Discrete outcomes are outcomes which are countable in direct comparison with integers.
For example, outcome no. 1, outcome no. 2, and so on. Continuous outcomes are outcomes
which are inﬁnitely many and noncountable. The outcomes are said to form a continuum. The
concept of an elementary event is not valid in this case. For example, consider choosing a
decimal number between 0 and l. The outcomes are continuous. Similarly, the value of a voltage or a current in an electric circuit is a continuous outcome. In quantifying events in terms of probability and using the results in applications, the following steps are undertaken: 1. The probabilities PM!) of some events 21, are estimated. 2. By using certain axioms, the probabilities P(B,) For some other events 31 are determined
from the probabilities P(Al). 3. Using the probabilities MB 1) , a prediction is made for the current application.
How to estimate the probabilities P(A,—) are discussed in the next section. The axioms to be used to determine NEE) are discussed in Section ??. CHAPTER 1 Probability Models in Electrical and Computer Engineering 1.1 Mathematical Models as Tools in Analysis and Design
1.2 Deterministic Models
1.3 Probability Models
Statistical Regularity
Properties of Relative F requemy
The Axiomalic Approach to a Theory of Probability
Building a Probability Model \DOOOUIPPN EXAMPLE. A fair die is thrown 2000 times. Predict how often the die shows up even. Solution: "Fair" means all outcomes are equally likely. We deduce that the probability of the eventA; of getting one of six possible numbers 1 thru 6 is 1X6. Half of the possible numbers are even, so the probability FOB) of the event 8 of the number being even is deduced to be 1f2. (These results are better explained in the following sections.) We predict that the number will be even about 1]?  2000: 100C times. ...
View
Full Document
 Spring '08
 GELFAND

Click to edit the document details