L¨ohr/Winter Winter term 2015/16
Exercises to the lecture Probability Theory II
Exercise sheet 1
Repetition
Exercise 1.1 (Normal distribution & central limit theorem).
(4 Points)Let f be the density of the standard normal distribution on
R, that is f (x) = (2π)
−12e
−12x2.
(a) Let X
nbe Poisson distributed with parameter n, n
∈N. Show for all x
∈Rthat
√
n
PX
n=
⌊n + x
√n
⌋ −→n→∞
f (x).
Hint:
Use Stirling’s formula.
(b) Let X be standard normally distributed. Calculate the density of Z = X
2.
Remark:The distribution of Z is called Γ(
12,
12)-distribution.
(c) Let S
n=
Pnk=1
X
k, where X
1, X
2, . . . are independent and uniformly distributed on
{−1, 1
}. Show that
n
lim
→∞PS
n>
√n log(n) = 0 and lim
n→∞P
|
S
n|>
log(√nn)= 1.
Hint:
Use the de Moivre-Laplace theorem (central limit theorem).
Exercise 1.2 (Conditional distribution).
(4 Points)(a) Let X = cos(2πU) and Y = sin(2πU ) for a random variable U , which is uniformly distributed on [0, 1]. Calculate
E(X
|Y ).
(b) Let X, Y be independent and uniformly distributed on [0, 1]. Calculate
E(X
|Z ) for Z = XY .
Hint:
Calculate at first the joint density of (X, Z ).
Exercise 1.3 (Law of large numbers for conditional expectations).
(4 Points)Let X
n, n
∈N, be independent and uniformly distributed on [0, 1]. Let Y be an arbitrary random variable, which can have arbitrary dependencies with the X
n. Show that the limit
n
lim
→∞1 n
n
X
k=1
E
(X
k |Y )
exists a.s. and calculate its value.
Please turn
Exercise 1.4 (Ergodic theorem).
(4 Points)Let U
1, U
2, . . . be independent and uniformly distributed on [0, 1]. Consider
X
n:=
n
Y
k=1
(U
k+ U
k+1)
n1and calculate lim
n→∞E(X
n).
Due Wed, 28.10. at the beginning of the exercise session
Probability Seminar:
20.10.: Robert Fitzner (Eindhoven University of Technology)
High-dimensional percolation
Abstract: Percolation is one of the simplest ways to define models in statistical physics and mathematics which displays a non-trivial critical behaviour. This model describes how a graph/lattice/network behaves under random removal of edges. In the last 40+ years, it has been an active field of research due to richness of the behaviour of the model and due to its numerous applications. We introduce the classical version of the model on the latticeZd, where percolation has the remarkable feature that it undergoes a sharp phase transition: There exists a critical valuepc =pc(d)∈(0,1) such that if you randomly retain a fraction of less thanpc edges (i.e., remove a fraction of more than 1−pcof all edges in i.i.d. manner), then the resulting graph consists of finite connected components only. Conversely, if you retain a fraction of more thanpc edges (i.e., randomly remove a smaller than 1−pc fraction of edges), then the resulting graph will contain exactly one infinite connected component. What happens at criticality, i.e., if we randomly retain exactly the critical fraction of pc edges is only understood for d= 2 (wherepc(2) = 0.5) and for d >10 (where pc(d)≈1/(2d−1)). In the talk, we review known results about percolation and give an idea of how to obtain results for d >10.
