Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
Theorem 1.14 Any function X from (IR,B) in (IR,B) that is
continuous is a r.v.
The probability of an event from IR induced by a r.v. is going
to be defined as the probability of the “original” events from Ω,
that is, the probability of a r.v. preserves the probabilities of the
original measurable space. This definition requires the measurability
property, since the “original” events must be in the initial
σ-algebra so that they have a probability.
Definition 1.40 (Probability induced by a r.v.)
Let (Ω,A,P) be a measure space and let B be the Borel σ-algebra
over IR. The probability induced by the r.v. X is a function
P X : B → IR, defined as
P X (B) = P(X −1 (B)), ∀B ∈ B.
Theorem 1.15 The probability induced by a r.v. X is a probability
function in (IR,B).
Example 1.18 For the probability function P 1 defined in Example
1.10 and the r.v. defined in Example 1.17 (a), the probability
induced by a r.v. X is described as follows. Let B ∈ B.
̌ If 0,1 ∈ B, then P 1X (B) = P 1 (X −1 (B)) = P 1 (Ω) = 1.
̌ If0 ∈ Bbut1 /∈ B,thenP 1X (B) = P 1 (X −1 (B)) = P 1 ({“tail”}) =
1/2.
̌ If1 ∈ Bbut0 /∈ B,thenP 1X (B) = P 1 (X −1 (B)) = P 1 ({“head”}) =
1/2.
̌ If 0,1 /∈ B, then P 1X (B) = P 1 (X −1 (B)) = P 1 (∅) = 0.
ISABEL MOLINA 38