probability measure or probability distribution is a finite measure whose total mass is equal to 1.
DEFINITION 1.7.– A probability or probability measure, or law of probability or distribution over a probability space (Ω,
– for any A ∈ , ℙ(A) ≥ 0,
– ℝ(Ω) = 1,
– for any sequence of pairwise disjoint events in , denoted by (An)n∈ℕ, we have
The triplet (Ω,
EXAMPLE 1.7.– Ω is endowed with the coarse σ-algebra
EXAMPLE 1.8.– Let Ω = [0, 1] and =
is a probability measure on (Ω,
EXAMPLE 1.9.– Let Ω be non-empty set such that card(Ω) < ∞, where card(Ω) denotes the cardinal of Ω, that is, the number of elements in Ω. Consider the mapping ℙ from
The mapping ℙ is then a probability on (Ω,
We will only review those properties of a probability that will be useful for this book.
PROPOSITION 1.2.– Let (Ω,
– If (An)n∈ℕ is increasing (for the inclusion), then,
– If (An)n∈ℕ is decreasing (for the inclusion), then,
We will now review the concept of independent events and σ-algebras.
DEFINITION 1.8.– Let (Ω,
– Two events, A and B, are independent if ℙ(A ∩ B) = ℙ(A) × ℙ(B).
– A family of events (Ai ∈ i, i ∈ I) is said to be mutually independent if for any finite family J ⊂ I, we have
– Two σ-algebras and are independent if for any A ∈ and B ∈ , A and B are independent.
– A family of sub-σ-algebra i ⊂ , i ∈ I is mutually independent if any family of events (Ai ∈ i, i ∈ I) is mutually independent.
EXAMPLE 1.10.– We roll a six-faced die and write
– A1 the event “the number obtained is even”; and
– A2 the event “the number obtained is a multiple of 3” .
The universe of possible outcomes is Ω = {1, 2, 3, 4, 5, 6} which has a finite number of elements and as all its elements have the same chance of occurring, we can endow it with the uniform probability . Since
we have
Therefore, A1 and A2 are two independent events.
EXAMPLE 1.11.– A coin is tossed twice. The following events are considered:
– A1 “Obtaining tails (T) on the first toss”;
– A2 “Obtaining heads (H) on the second toss”; and
– A3 “Obtaining the same face on both tosses”.
The universe of possible outcomes is
which has four elements, and as all elements have the same chance of occurring, it can be endowed with uniform probability. Since
we have
and
