Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>1.</strong>8. RANDOM VARIABLES<br />
Theorem <strong>1.</strong>14 Any function X from (IR,B) in (IR,B) that is<br />
continuous is a r.v.<br />
The probability of an event from IR induced by a r.v. is going<br />
to be defined as the probability of the “original” events from Ω,<br />
that is, the probability of a r.v. preserves the probabilities of the<br />
original measurable space. This definition requires the measurability<br />
property, since the “original” events must be in the initial<br />
σ-algebra so that they have a probability.<br />
Definition <strong>1.</strong>40 (<strong>Probability</strong> induced by a r.v.)<br />
Let (Ω,A,P) be a measure space <strong>and</strong> let B be the Borel σ-algebra<br />
over IR. The probability induced by the r.v. X is a function<br />
P X : B → IR, defined as<br />
P X (B) = P(X −1 (B)), ∀B ∈ B.<br />
Theorem <strong>1.</strong>15 The probability induced by a r.v. X is a probability<br />
function in (IR,B).<br />
Example <strong>1.</strong>18 For the probability function P 1 defined in Example<br />
<strong>1.</strong>10 <strong>and</strong> the r.v. defined in Example <strong>1.</strong>17 (a), the probability<br />
induced by a r.v. X is described as follows. Let B ∈ B.<br />
̌ If 0,1 ∈ B, then P 1X (B) = P 1 (X −1 (B)) = P 1 (Ω) = <strong>1.</strong><br />
̌ If0 ∈ Bbut1 /∈ B,thenP 1X (B) = P 1 (X −1 (B)) = P 1 ({“tail”}) =<br />
1/2.<br />
̌ If1 ∈ Bbut0 /∈ B,thenP 1X (B) = P 1 (X −1 (B)) = P 1 ({“head”}) =<br />
1/2.<br />
̌ If 0,1 /∈ B, then P 1X (B) = P 1 (X −1 (B)) = P 1 (∅) = 0.<br />
ISABEL MOLINA 38