Download event A

Dr. Ahmed Abdelwahab
Introduction for
EE420
Probability Theory
Probability theory is rooted in phenomena that can be modeled
by an experiment with an outcome that is subject to chance.
Moreover, if the experiment is repeated, the outcome can differ
because of the influence of an underlying random phenomenon or
chance mechanism. Such an experiment is referred to as a
random experiment.
The following three features describe the random experiment:
1. The experiment is repeatable under identical conditions.
2. On any trial of the experiment, the outcome is unpredictable.
3. For a large number of trials of the experiment, the outcomes
exhibit statistical regularity; that is, a definite average pattern of
outcomes is observed if the experiment is repeated a large
number of times.
Let event A denote one of the possible outcomes of a
random experiment. For example, in the coin-tossing
experiment, event A may represent "heads." Suppose
that in n trials of the experiment, event A occurs N(A)
times. We may then assign the ratio N(A)/n to the
event A. This ratio is called the relative frequency of
the event A. Clearly, the relative frequency is a
nonnegative real number less than or equal to one.
That is to say,
and the probability of
event A is defined as
The total of all possible outcomes of the random
experiment is called the sample space, which is
denoted by S. An event corresponds to either a single
sample point or a set of sample points. In particular,
the entire sample space S is called the sure event; the
null set ϕ is called the null or impossible event; and a
single sample point is called an elementary event.
A probability system consists of the following three aspects:
1. A sample space S of elementary events (outcomes).
2. A class ξ of events that are subsets of S.
3. A probability measure P(.) assigned to each event A in the
class ξ, which has the following properties:
(i)
P(S) = 1
(ii) 0 ≤ P(A) ≤ 1
(iii) If A + B is the union of two mutually exclusive events in
the class ξ, then P(A + B) = P(A) + P(B)
Properties (i), (ii), and (iii) are known as the axioms of
probability. Axioms (i), (ii), and (iii) constitute an implicit
definition of probability.
Joint Probability
P(not a) = 1 − P(a).
Union of a & b = P(a or b) = P(a+b)
= P(a)+P(b) − P(a and b).
We will often denote P(a and b) as the intersection
(joint probability) of a & b by P(a,b) or simply P(ab), i.e.
P(ab) = P(a)+P(b) − P(a+b).
If P(ab) = 0, we say a and b are mutually exclusive (or disjoint).
Conditional probability
P(a|b) is the probability of a, given that we know b
P(a|b) is called Conditional probability of a given b.
The joint probability of both a and b is given by:
P(a, b) = P(a|b) P(b) and since P(a,b) = P(b,a),
we have Bayes’Theorem :
P(a,b) = P(a|b)P(b) = P(b,a) = P(b|a)P(a)
Statistical Independence
If two events a and b are such that P(a|b) = P(a),
then, the events a and b are statistically
independent (SI).
Note that from Bayes’Theorem, also, P(b|a) = P(b),
therefore, P(ab) = P(a|b)P(b) = P(a)P(b).
This last equation is often taken as the
definition of statistical independence.
Statistics of Continuous Random
Variable
Gaussian Distribution
The random variable Y has a Gaussian distribution if its probability
density function has the form:
where µY is the mean and σY2 is
the variance of the random variable Y.
Normalized Gaussian distribution has
µY =0 and σY2 =1 as shown in the figure.