Page 16 - Elements of Distribution Theory

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052184472Xc01 CUNY148/Severini May 24, 2005 17:52

2 Properties of Probability Distributions

(P3) If A 1 , A 2 ,..., are disjoint subsets of , then

∞ ∞

P A n = P(A n ).
n=1 n=1
Note that (P3) implies (P2); however, (P3), which is concerned with an infinite sequence
of events, is of a different nature than (P2) and it is useful to consider them separately.
There are a number of straightforward consequences of (P1)–(P3). For instance, P(∅) = 0,
c
c
if A denotes the complement of A, then P(A ) = 1 − P(A), and, for A 1 , A 2 not necessarily
disjoint,
P(A 1 ∪ A 2 ) = P(A 1 ) + P(A 2 ) − P(A 1 ∩ A 2 ).

Example 1.1 (Sampling from a finite population). Suppose that is a finite set and that,
for each ω ∈ ,

P({ω}) = c
for some constant c. Clearly, c = 1/| | where | | denotes the cardinality of .
Let A denote a subset of . Then
|A|
P(A) = .
| |
Thus, the problem of determining P(A)is essentially the problem of counting the number
of elements in A and .

Example 1.2 (Bernoulli trials). Let
n
={x ∈ R : x = (x 1 ,..., x n ), x j = 0or 1, j = 1,..., n}
so that an element of is a vector of ones and zeros. For ω = (x 1 ,..., x n ) ∈ , take
n

x j
P(ω) = θ (1 − θ) 1−x j
j=1
where 0 <θ < 1isagiven constant.

Example 1.3 (Uniform distribution). Suppose that = (0, 1) and suppose that the prob-
ability of any interval in is the length of the interval. More generally, we may take the
probability of a subset A of to be

P(A) = dx.
A
Ideally, P is defined on the set of all subsets of . Unfortunately, it is not generally
possible to do so and still have properties (P1)–(P3) be satisfied. Instead P is defined only
on a set F of subsets of ;if A ⊂ is not in F, then P(A)is not defined. The sets in F
are said to be measurable. The triple ( , F, P) is called a probability space; for example,
we might refer to a random variable X defined on some probability space.
Clearly for such an approach to probability theory to be useful for applications, the set
F must contain all subsets of of practical interest. For instance, when is a countable
set, F may be taken to be the set of all subsets of . When may be taken to be a