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 />
Definition <strong>1.</strong>43 Ar.v. X issaidtobediscrete (absolutely continuous)<br />
if <strong>and</strong> only if its d.f. F X is discrete (absolutely continuous).<br />
Remark <strong>1.</strong>8 It holds that:<br />
(a) a discrete r.v. takes a finite or countable number of values.<br />
(b) a continuous r.v. takes infinite number of values, <strong>and</strong> the<br />
probability of single values are zero.<br />
Definition <strong>1.</strong>44 (Support of a r.v.)<br />
(a) If X is a discrete r.v., we define the support of X as<br />
D X := {x ∈ IR : P X {x} > 0}.<br />
(b) If X continuous, the support is defined as<br />
D X := {x ∈ IR : f X (x) > 0}.<br />
∑<br />
Observe that for a discrete r.v., P X {x} = 1 <strong>and</strong> D X is finite<br />
x∈D X<br />
or countable.<br />
Example <strong>1.</strong>20 From the r<strong>and</strong>om variables introduced in Example<br />
<strong>1.</strong>17, those defined in (a) <strong>and</strong> (b) are discrete, along with X 2<br />
form (c).<br />
For a discrete r.v. X, we can define a function the gives the<br />
probabilities of single points.<br />
Definition <strong>1.</strong>45 Theprobability mass function (p.m.f)ofadiscrete<br />
r.v. X is the function p X : IR → IR such that<br />
p X (x) = P X ({x}), ∀x ∈ IR.<br />
ISABEL MOLINA 40