Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
(b) For the measurable space (Ω,P(Ω)), where Ω = IN ∪ {0},
since Ω ⊂ IR, a trivial r.v. is X 1 (ω) = ω. It is a r.v. since for
any B ∈ B,
X −1
1 (B) = {ω ∈ IN ∪{0} : X 1(ω) = ω ∈ B}
is the set of natural numbers (including zero) that are containedinB.
Butanycountablesetofnaturalnumbersbelongs
to P(Ω), since this σ-algebra contains all subsets of IN ∪{0}.
Therefore, X 1 =“Number of traffic accidents in a minute in
Spain” is a r.v.
Another r.v. could be
X 2 (ω) =
{ 1 if ω ∈ IN;
0 if ω = 0.
Again, X 2 is a r.v. since for each B ∈ B,
̌ If 0,1 ∈ B, then X2 −1 (B) = Ω ∈ P(Ω).
̌ If 1 ∈ B but 0 /∈ B, then X2 −1 (B) = IN ∈ P(Ω).
̌ If 0 ∈ B but 1 /∈ B, then X2 −1 (B) = {0} ∈ P(Ω).
̌ If 0,1 /∈ B, then X2 −1 (B) = ∅ ∈ P(Ω).
(c) As in previous example, for the measurable space (Ω,B Ω ),
where Ω = [m,∞), a possible r.v. is X 1 (ω) = ω, since for
each B ∈ B, we have
X −1
1 (B) = {ω ∈ [a,∞) : X 1(ω) = ω ∈ B} = [a,∞)∩B ∈ B Ω .
Another r.v. would be the indicator of less than 65 kgs., given
by
{ 1 if ω ≥ 65,
X 2 (ω) =
0 if ω < 65.
ISABEL MOLINA 37