Slides Chapter 1. Measure Theory and Probability

1.4. PROBABILITY MEASURES
By axiom (ii), a probability function is a measure for which the
measure of the sample space Ω is 1. The triplet (Ω,A,P), where
P is a probability P, is called probability space.
Example 1.10 Fortheexperiment(a)describedinExample1.8,
with the measurable space (Ω,A), where
define
Ω = {“head”,“tail”}, A = {∅,{“head”},{“tail”},Ω},
P 1 (∅) = 0, P 1 ({“head”}) = p, P 1 ({“tail”}) = 1−p, P 1 (Ω) = 1,
where p ∈ [0,1]. This function verifies the axioms of Kolmogoroff.
Example 1.11 For the experiment (b) described is Example 1.8,
with the measurable space (Ω,P(Ω)), define:
• For the elementary events, the probability is
P({0}) = 0.131, P({1}) = 0.272, P({2}) = 0.27, P({3}) = 0.183,
P({4}) = 0.09, P({5}) = 0.012, P({6}) = 0.00095, P({7}) = 0.00005,
P(∅) = 0, P({i}) = 0, ∀i ≥ 8.
• For other events, the probability is defined as the sum of the
probabilities of the elementary events contains in that event,
that is, if A = {a 1 ,...,a n }, where a i ∈ Ω are the elementary
events, the probability of A is
n∑
P(A) = P({a i }).
i=1
This function verifies the axioms of Kolmogoroff.
Proposition 1.4 (Properties of the probability)
The following properties are consequence of the axioms of Kolmogoroff:
ISABEL MOLINA 14