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 />
Summarizing, the probability induced by X is<br />
⎧<br />
⎨ 0, if 0,1 /∈ B;<br />
P 1X (B) = 1/2, if 0 or 1 are in B;<br />
⎩<br />
1, if 0,1 ∈ B.<br />
In particular, we obtain the following probabilities<br />
̌ P 1X ({0}) = P 1 (X = 0) = 1/2.<br />
̌ P 1X ((−∞,0]) = P 1 (X ≤ 0) = 1/2.<br />
̌ P 1X ((0,1]) = P 1 (0 < X ≤ 1) = 1/2.<br />
Example <strong>1.</strong>19 For the probability function P introduced in Example<br />
<strong>1.</strong>11 <strong>and</strong> the r.v. X 1 defined in Example <strong>1.</strong>17 (b), the probability<br />
induced by the r.v. X 1 is described as follows. Let B ∈ B<br />
such that IN ∩B = {a 1 ,a 2 ,...,a p }.<br />
P X1 (B) = P(X −1<br />
1 (B)) = P 1((IN∪{0})∩B) = P({a 1 ,a 2 ,...,a p }) =<br />
Definition <strong>1.</strong>41 (Degenerate r.v.)<br />
A r.v. X is said to be degenerate at a point c ∈ IR if <strong>and</strong> only if<br />
p∑<br />
P({a i<br />
i=1<br />
P(X = c) = P X ({c}) = <strong>1.</strong><br />
Since P X is a probability function, there is a distribution function<br />
that summarizes its values.<br />
Definition <strong>1.</strong>42 (Distribucion function)<br />
The distribucion function (d.f.) of a r.v. X is defined as the<br />
function F X : IR → [0,1] with<br />
F X (x) = P X (−∞,x], ∀x ∈ IR.<br />
ISABEL MOLINA 39