Page 20 - Probability, Random Variables and Random Processes

PROBABILITY [CHAP 1

Since each subset of S can be uniquely characterized by an element in the above Cartesian product, we
obtain the number of elements in Q by
'
n(Q) = n(S,)n(S,) - - . n(S,) = 2"
where n(Si) = number of elements in Si = 2.
An alternative way of finding n(Q) is by the following summation:
" nl
n(Ql= (y) =
i=O i=o i!(n - i)!
The proof that the last sum is equal to 2" is not easy.

ALGEBRA OF SETS
1.9. Consider the experiment of Example 1.2. We define the events
A = {k: k is odd)
B={k:4<k17)
C = {k: 1 5 k 5 10)
where k is the number of tosses required until the first H (head) appears. Determine the events A,
B,C,Au B,BuC,An B,AnC,BnC,andAn B.
= (k: k is even) = (2, 4, 6, . . .)
B = {k: k = 1, 2, 3 or k 2 8)
C= (k: kr 11)
A u B = {k: k is odd or k = 4, 6)
BuC=C
A n B = (5, 7)
A n C = {I, 3, 5, 7, 9)
BnC=B
A n B = (4, 6)

1.10. The sample space of an experiment is the real line expressed as

(a) Consider the events
A, = {v: 0 S v < $1
A, = {v: f 5 V < $1

Determine the events

U Ai and A,
i= 1 i= 1
(b) Consider the events
B, = {v: v 5 1
B, = {v: v < 3)