Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
We will also use the notation p(X = x) = p X (x).
The probability function induced by a discrete r.v. X, P X , is
completely determined by the distribution function F X or by the
mass function p X . Thus, in the following, when we speak about
the “distribution” of a discrete r.v. X, we could be referring either
to the probability function induced by X, P X , the distribution
function F X , or the mass function p X .
Example 1.21 The r.v. X 1 : “Weight of a randomly selected
Spanish woman aged within 20 and 40”, defined in Example 1.17
(c), is continuous.
Definition 1.46 The probability density function (p.d.f.) of X
is a function f X : IR → IR defined as
{ 0 if x ∈ S;
f X (x) =
F X ′ (x) if x /∈ S.
It is named probability density function of x because it gives the
density of probability of an infinitesimal interval centered in x.
The same as in the discrete case, the probabilities of a continuous
r.v. X are determined either by P X , F X or the p.d.f. f X .
Again, the “distribution” of a r.v., could be referring to any of
these functions.
Randomvariables, asmeasurable functions, inheritallthepropertiesofmeasurablefunctions.
Furthermore, wewillbeabletocalculate
Lebesgue integrals of measurable functions of r.v.’s using as
measure their induced probability functions. This will be possible
due to the following theorem.
Theorem 1.16 (Theorem of change of integrationspace)
Let X be a r.v. from (Ω,A,P) in (IR,B) an g another r.v. from
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