Читать книгу Martingales and Financial Mathematics in Discrete Time онлайн

14 страница из 24

 – 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

12

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