Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
over IR. A random variable (r.v.) is a function X : Ω → IR that
is measurable, that is,
∀B ∈ B, X −1 (B) ∈ A,
where X −1 (B) := {ω ∈ Ω : X(ω) ∈ B}.
Remark 1.7 Observe that:
(a) a r.v. X is simply a measurable function in IR. The name
random variable stems from the fact the result of the random
experiment ω ∈ Ω is random, and then the observed value of
the r.v., X(ω), is also random.
(b) the measurability property of the r.v. will allow transferring
probabilities of events A ∈ A to probabilities of Borel sets
I ∈ B, where I is the image of A through X.
Example 1.17 For the experiments introduced in Example 1.8,
the following are random variables:
(a) Forthemeasurablespace(Ω,A)withsamplespaceΩ = {“head”,“tail”}
and σ-algebra A = {∅,{“head”},{“tail”},Ω}, a random
variable is: { 1 if ω = “head”,
X(ω) =
0 if ω = “tail”;
This variable counts the number of heads when tossing a coin.
In fact, it is a random variable, since for any event from the
final space B ∈ B, we have
̌ If 0,1 ∈ B, then X −1 (B) = Ω ∈ A.
̌ If 0 ∈ B but 1 /∈ B, then X −1 (B) = {“tail”} ∈ A.
̌ If 1 ∈ B but 0 /∈ B, then X −1 (B) = {“head”} ∈ A.
̌ If 0,1 /∈ B, then X −1 (B) = ∅ ∈ A.
ISABEL MOLINA 36