48
D. Krpelík and T. Basu
For a set E ∈ A X , the probability that the RV X obtains a value in E is
P (X ∈ E) = P (X
−1 (E)) = P ({ω : X(ω) ∈ E})
Δ
= P X (E).
(2.7)
Each RV induces its own probability field (Ω X , A X , P X ), where P X = P ◦ X −1 .
We will use notation X ∼ P X for denoting that P X is the (induced) measure of X.
For the derived quantities, say Z = f (X, Y ), we may proceed analogically.
Furthermore, the situation is simplified if the sample spaces Ω and Ω X are the
real lines and the mapping defining the new RV is monotone.
Theorem 2.1 Let X : (R, B, P ) → (R, B) be a strictly increasing RV and
[a, b] ⊂ R an interval. Because there exists unique classical inverse X −1 of
X, which is also an increasing function, we can express the distribution of X
as
P X ([a, b]) = P (X ∈ [a, b]) = P (X < b) − P (X < a)
= P ([X
−1 (a), X
−1 (b)]).
And similarly for a decreasing X, where the interval for the pre-image is given
by swapping the bounds, i.e. [X −1 (b), X −1 (a)].
See that Theorem 2.1 was exploited in the examples presented in Sects. 2.2.3
and 2.2.4.
In probability theory, special attention is given to the CDFs, which represent
the probability that a real-valued RV obtains a value smaller than the argument.
A CDF F X : R → [0, 1] represents, for each a ∈ R, the probability of event
{X ∈ [−∞, a]}. The probability of all the other events in the Borel algebra on the
real line, B, can be derived through the axioms of probability measures. Therefore,
CDF uniquely represents the whole probability measure, which would be intractable
to work with otherwise.
In special cases, the CDF of a derived quantity can be easily derived from
Theorem 2.1. For increasing functions f , the CDF of an extended RV Y = f (X)
can be calculated as
F Y (y) = P (f (X) < y) = F X (f
−1 (y)).
Similarly, for the case of a decreasing function f , where F Y (y) = 1 − F X (f −1 (y)).
Compared with the analysis on the real line, it is convenient to also introduce
some summaries of the RVs. This can be done using the expected value functional
(E : L(Ω) → R), where L is the space of all real RVs on the sample space Ω. For
that, we need to equip the underlying probability field with an integration operator
(e.g. Lebesgue–Stieltjes integral).
D. Krpelík and T. Basu
For a set E ∈ A X , the probability that the RV X obtains a value in E is
P (X ∈ E) = P (X
−1 (E)) = P ({ω : X(ω) ∈ E})
Δ
= P X (E).
(2.7)
Each RV induces its own probability field (Ω X , A X , P X ), where P X = P ◦ X −1 .
We will use notation X ∼ P X for denoting that P X is the (induced) measure of X.
For the derived quantities, say Z = f (X, Y ), we may proceed analogically.
Furthermore, the situation is simplified if the sample spaces Ω and Ω X are the
real lines and the mapping defining the new RV is monotone.
Theorem 2.1 Let X : (R, B, P ) → (R, B) be a strictly increasing RV and
[a, b] ⊂ R an interval. Because there exists unique classical inverse X −1 of
X, which is also an increasing function, we can express the distribution of X
as
P X ([a, b]) = P (X ∈ [a, b]) = P (X < b) − P (X < a)
= P ([X
−1 (a), X
−1 (b)]).
And similarly for a decreasing X, where the interval for the pre-image is given
by swapping the bounds, i.e. [X −1 (b), X −1 (a)].
See that Theorem 2.1 was exploited in the examples presented in Sects. 2.2.3
and 2.2.4.
In probability theory, special attention is given to the CDFs, which represent
the probability that a real-valued RV obtains a value smaller than the argument.
A CDF F X : R → [0, 1] represents, for each a ∈ R, the probability of event
{X ∈ [−∞, a]}. The probability of all the other events in the Borel algebra on the
real line, B, can be derived through the axioms of probability measures. Therefore,
CDF uniquely represents the whole probability measure, which would be intractable
to work with otherwise.
In special cases, the CDF of a derived quantity can be easily derived from
Theorem 2.1. For increasing functions f , the CDF of an extended RV Y = f (X)
can be calculated as
F Y (y) = P (f (X) < y) = F X (f
−1 (y)).
Similarly, for the case of a decreasing function f , where F Y (y) = 1 − F X (f −1 (y)).
Compared with the analysis on the real line, it is convenient to also introduce
some summaries of the RVs. This can be done using the expected value functional
(E : L(Ω) → R), where L is the space of all real RVs on the sample space Ω. For
that, we need to equip the underlying probability field with an integration operator
(e.g. Lebesgue–Stieltjes integral).