Universität zu Köln
Winter semester 2016 / 2017
Institut für Mathematik
Lecturer: Prof. Dr. A. Drewitz
Assistant: L. Schmitz
To be handed in on December 21st before exercise class.
8th Homework Probability Theory II
(Regular conditional probability and distribution)
To abbreviate notation, in every exercise we assume the random variables to
be defined on a probability space (Ω, F, P).
Exercise 8.1
(3 points)
Let T = [0, ∞). We say that a stochastic process (Xt ) has independent increments, if for every s, t ∈ T the increment Xt+s −Xt is independent of Ft (where
(Ft ) denotes the filtration generated by (Xt )).
Show that every stochastic process with independent increments has the Markov property.
Exercise 8.2
(6 points)
Let (Ω, F, P) = ([0, 1], B ([0, 1]), λ ([0, 1])) and f : F → Ω defined via
(
sup(F ), F 6= ∅,
f (F ) :=
0,
F = ∅,
1
1
where sup(F ) = inf{y ∈ Ω : y ≥ x for all x ∈ F } is the usual supremum for
F 6= ∅. Let
µ̃ : Ω × F → R, µ̃(ω, F ) := 1F (ω) + 1f (F ) (ω).
Show that
a) for every F ∈ F the mapping ω 7→ µ̃(ω, F ) defines a version of the
conditional probability P(F | F);
(2 p.)
b) there exists no null set N ∈ F, such that for all ω ∈ N c the mapping
F 7→ µ̃(ω, F ) is a probability measure on F;
(3 p.)
c) a regular conditional probability given F exists and give an explicit expression for it.
(1 p.)
Exercise 8.3
(6 points)
2
2
2
Let (Ω, F, P) = (R , B(R ), P ), where P := f · λ with
f : R2 → R, f (x, y) =
1 − x2 +y2
e 2 .
2π
Let X ∈ L1 . Give a version of E [X |G] for the symmetry set
a) G = Gaxis = {B ∈ F : (x, y) ∈ B ⇔ (−x, y) ∈ B};
(3 p.)
b) G = Grot = {B ∈ F : (x, y) ∈ B and x2 + y 2 = u2 + v 2 ⇒ (u, v) ∈ B}.
(3 p.)
Don’t forget to prove that G is a sub-σ-algebra of F.
R
Hint for b): Write B X(x, y)f (x, y)d(x, y) w.r.t. polar coordinates.
Exercise 8.4
(0 points)
Let X, Y be real random variables, such that the random vector (X, Y ) has
probability density f w.r.t. the two-dimensional Lebesgue measure λ2 on B 2 .
Let µ̃ be a fixed probability measure on B and define
(R
f (y|x) dy, if fX (x) ∈ (0, ∞),
B Y |X
µ(x, B) :=
µ̃(B),
else,
B ∈ B, where we use the same notation as in Example 3.1.7. a).
a) Show that µ(x, ·) indeed defines the regular conditional distribution of Y
given X.
b) Let (X, Y ) be uniformly distributed on the disc
B := (x, y) ∈ R2 : x2 + y 2 ≤ 1 .
Calculate the conditional distribution of Y given X.
If you think there are mistakes or inconsistencies, don’t hesitate to contact me:
[email protected]
Remark: Please just hand in solely exercises with positive rating. If you need
more than one sheet, please clip together the papers and write your group, your
name and the exercise on top of every sheet.
It is possible to hand in solutions in groups of at most two persons.
Total: 15
2