Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
Definition 1.43 Ar.v. X issaidtobediscrete (absolutely continuous)
if and only if its d.f. F X is discrete (absolutely continuous).
Remark 1.8 It holds that:
(a) a discrete r.v. takes a finite or countable number of values.
(b) a continuous r.v. takes infinite number of values, and the
probability of single values are zero.
Definition 1.44 (Support of a r.v.)
(a) If X is a discrete r.v., we define the support of X as
D X := {x ∈ IR : P X {x} > 0}.
(b) If X continuous, the support is defined as
D X := {x ∈ IR : f X (x) > 0}.
∑
Observe that for a discrete r.v., P X {x} = 1 and D X is finite
x∈D X
or countable.
Example 1.20 From the random variables introduced in Example
1.17, those defined in (a) and (b) are discrete, along with X 2
form (c).
For a discrete r.v. X, we can define a function the gives the
probabilities of single points.
Definition 1.45 Theprobability mass function (p.m.f)ofadiscrete
r.v. X is the function p X : IR → IR such that
p X (x) = P X ({x}), ∀x ∈ IR.
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