Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>8. RANDOM VARIABLES<br />
over IR. A r<strong>and</strong>om variable (r.v.) is a function X : Ω → IR that<br />
is measurable, that is,<br />
∀B ∈ B, X −1 (B) ∈ A,<br />
where X −1 (B) := {ω ∈ Ω : X(ω) ∈ B}.<br />
Remark <strong>1.</strong>7 Observe that:<br />
(a) a r.v. X is simply a measurable function in IR. The name<br />
r<strong>and</strong>om variable stems from the fact the result of the r<strong>and</strong>om<br />
experiment ω ∈ Ω is r<strong>and</strong>om, <strong>and</strong> then the observed value of<br />
the r.v., X(ω), is also r<strong>and</strong>om.<br />
(b) the measurability property of the r.v. will allow transferring<br />
probabilities of events A ∈ A to probabilities of Borel sets<br />
I ∈ B, where I is the image of A through X.<br />
Example <strong>1.</strong>17 For the experiments introduced in Example <strong>1.</strong>8,<br />
the following are r<strong>and</strong>om variables:<br />
(a) Forthemeasurablespace(Ω,A)withsamplespaceΩ = {“head”,“tail”}<br />
<strong>and</strong> σ-algebra A = {∅,{“head”},{“tail”},Ω}, a r<strong>and</strong>om<br />
variable is: { 1 if ω = “head”,<br />
X(ω) =<br />
0 if ω = “tail”;<br />
This variable counts the number of heads when tossing a coin.<br />
In fact, it is a r<strong>and</strong>om variable, since for any event from the<br />
final space B ∈ B, we have<br />
̌ If 0,1 ∈ B, then X −1 (B) = Ω ∈ A.<br />
̌ If 0 ∈ B but 1 /∈ B, then X −1 (B) = {“tail”} ∈ A.<br />
̌ If 1 ∈ B but 0 /∈ B, then X −1 (B) = {“head”} ∈ A.<br />
̌ If 0,1 /∈ B, then X −1 (B) = ∅ ∈ A.<br />
ISABEL MOLINA 36