Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
(b) If f is continuous in the point x, then F is differentiable in x
and F ′ (x) = f(x).
(c) P F ({x}) = 0, ∀x ∈ IR.
(d) P F (a,b) = P F (a,b] = P F [a,b) = P F [a,b] = ∫ b
a
with a < b.
(e) P F (B) = ∫ Bf(t)dt,∀B ∈ B.
Remark 1.6 Note that:
(1) Not all continuous d.f.’s are absolutely continuous.
f(t)dt, ∀a,b
(2) Anothertypeofd.f’sarethosecalledsingular d.f.’s, whichare
continuous. We will not study them.
Proposition 1.18 Let F 1 ,F 2 be d.f.’s and λ ∈ [0,1]. Then,
F = λF 1 +(1−λ)F 2 is a d.f.
Definition 1.38 (Mixed d.f.)
A d.f. is said to be mixed if and only if there is a discrete d.f.
F 1 , an absolutely continuous d.f. F 2 and λ ∈ [0,1] such that
F = λF 1 +(1−λ)F 2 .
1.8 Random variables
A random variable transforms the elements of the sample space
Ω into real numbers (elements from IR), preserving the σ-algebra
structure of the initial events.
Definition 1.39 Let (Ω,A) be a measurable space. Consider
also the measurable space (IR,B), where B is the Borel σ-algebra
ISABEL MOLINA 35