L¨ohr/Winter Winter term 2015/16
Exercises to the lecture Probability Theory II
Exercise sheet 5
Separating classes & Characteristic functions
Exercise 5.1 (Cb(E) is separating in M1(E)). (4 Points) Let (E, d) be a metric space. Show thatCb(E) is separating inM1(E) (without using theorems from the lecture which used this property in their proof).
Exercise 5.2(Moment problem). (4 Points)
(a) Let (E, d) be a compact, metric space,F ⊆ Cb(E) separating points and stable under multiplications. Let Xn, X random variables with values inE and
E f(Xn)
−→
n→∞ E f(X)
∀f ∈ F.
Show that (Xn)n∈N converges in distribution toX.
(b) Let (Xn)n∈N be a sequence of [0,1]-valued random variables, and mn,k := E(Xnk).
Furthermore, assume that limn→∞mn,k =mk for suitablemk∈R. Show that there is a random variable X, such that
E(Xk) = mk and Xn ⇒ X.
Hint: Use Prohorov’s Theorem and (a).
Exercise 5.3(Examples of characteristic functions). (4 Points) Letp∈]0,1[, n∈N,λ > 0. Calculate the characteristic functions of the following distribu- tions onR:
(a)δ3, (b) binomial distribution Bin(n, p), (c) exponential distributionEλ.
Exercise 5.4. (4 Points)
(a) Let λ >0. Show thatφ(t) := exp λ(eit−1)
is the characteristic function of a Poisson distribution with parameter λ.
(b) Considerψ:R→C,ψ(t) := exp 2i·t−42·t2+eit−1
. Show that there is a random variable X with characteristic function ψ. Furthermore, calculate Var(X), and show that P {X ≥3}
≥ 14.
Due Wed, 25.11. at the beginning of the exercise session
Probability Seminar:
17.11.: Daniel Zivkovic (LMU Munich) .
24.11.: Stefan H¨afner (University of Duisburg-Essen) . Tue, 16:15 – 17:15in WSC-S-U-3.03
