Slides Chapter 1. Measure Theory and Probability

1.8. RANDOM VARIABLES
Summarizing, the probability induced by X is
⎧
⎨ 0, if 0,1 /∈ B;
P 1X (B) = 1/2, if 0 or 1 are in B;
⎩
1, if 0,1 ∈ B.
In particular, we obtain the following probabilities
̌ P 1X ({0}) = P 1 (X = 0) = 1/2.
̌ P 1X ((−∞,0]) = P 1 (X ≤ 0) = 1/2.
̌ P 1X ((0,1]) = P 1 (0 < X ≤ 1) = 1/2.
Example 1.19 For the probability function P introduced in Example
1.11 and the r.v. X 1 defined in Example 1.17 (b), the probability
induced by the r.v. X 1 is described as follows. Let B ∈ B
such that IN ∩B = {a 1 ,a 2 ,...,a p }.
P X1 (B) = P(X −1
1 (B)) = P 1((IN∪{0})∩B) = P({a 1 ,a 2 ,...,a p }) =
Definition 1.41 (Degenerate r.v.)
A r.v. X is said to be degenerate at a point c ∈ IR if and only if
p∑
P({a i
i=1
P(X = c) = P X ({c}) = 1.
Since P X is a probability function, there is a distribution function
that summarizes its values.
Definition 1.42 (Distribucion function)
The distribucion function (d.f.) of a r.v. X is defined as the
function F X : IR → [0,1] with
F X (x) = P X (−∞,x], ∀x ∈ IR.
ISABEL MOLINA 39