Page 10 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

sample points (as in Example 1.1) or countably infinite sample points (as in Example 1.2). A set is called countable
if its elements can be placed in a one-to-one correspondence with the positive integers. A sample space S is said
to be continuous if the sample points constitute a continuum (as in Example 1.3).

C. Events:

Since we have identified a sample space S as the set of all possible outcomes of a random experi-
ment, we will review some set notations in the following.
If C is an element of S (or belongs to S), then we write

If S is not an element of S (or does not belong to S), then we write
us
A set A is called a subset of B, denoted by
AcB
if every element of A is also an element of B. Any subset of the sample space S is called an event. A
sample point of S is often referred to as an elementary event. Note that the sample space S is the
subset of itself, that is, S c S. Since S is the set of all possible outcomes, it is often called the certain
event.

EXAMPLE 1.4 Consider the experiment of Example 1.2. Let A be the event that the number of tosses required
until the first head appears is even. Let B be the event that the number of tosses required until the first head
appears is odd. Let C be the event that the number of tosses required until the first head appears is less than 5.
Express events A, B, and C.

1.3 ALGEBRA OF SETS
A. Set Operations:
I. Equality:
Two sets A and B are equal, denoted A = B, if and only if A c B and B c A.

2. Complementation :
Suppose A c S. The complement of set A, denoted A, is the set containing all elements in S but
not in A.
A= {C: C: E Sand $ A)
3. Union:
The union of sets A and B, denoted A u B, is the set containing all elements in either A or B or
both.

4. Intersection:
The intersection of sets A and B, denoted A n B, is the set containing all elements in both A
and B.