Aalto University
Department of Mathematics and Systems Analysis
MS-E1600 - Probability theory, 2017/III
Problem set 3
K Kytölä & J Karjalainen
3 Stochastic dependence and independence
Exercise session: Thu 19.1. at 14-16
Solutions due: Mon 23.1. at 10
Reminders: Throughout, let (Ω, F, P) denote a probability space.
Independence. A collection (Gj )j∈J of σ-algebras Gj on Ω is independent
iff for any m ∈ N and all distinct
j1 , . . . , jm ∈ J and all sets Ej1 ∈ Gj1 , . . . , Ejm ∈ Gjm we have P Ej1 ∩· · ·∩Ejm = P[Ej1 ] · · · P[Ejm ]. A collection (Xj )j∈J of random variables is independent iff the collection σ(Xj ) j∈J of σ-algebras (generated
by the corresponding random variables) is independent. A collection (Ej )j∈J of events is independent
iff the collection IEj j∈J of indicator random variables (of corresponding events) is independent. There
are, of course, other more practical sufficient conditions for independence that you may use.
Tail σ-algebra. For X1 , X2 , X3 , . . . : Ω → S 0 random variables, the tail σ-algebra is defined as T∞ =
T
m∈N σ(Xk : k ≥ m), where σ(Xk : k ≥ m) is the σ-algebra generated by the collection {Xm , Xm+1 , Xm+2 , . . .}.
3.1 Independent pairs. Let X and Y be R-valued random variables defined on a common
probability space (Ω, F, P).
(a) Show that if X and Y are independent, then f (X) and g(Y ) are independent
for any Borel-measurable functions f, g : R → R.
(b) Show that if f (X) and g(Y ) are independent for all Borel-measurable functions
f, g : R → R, then X and Y are independent.
(c) Assume that P X +Y = 42 = 1. Is it possible that X and Y are independent?
3.2 Two independent geometrically distributed numbers. Let X, Y : Ω → N be two independent random variables with P[X = j] = P[Y = j] = 21j for all j ∈ N = {1, 2, . . .}.
(a) Show that P[Y > n] =
1
2n
for any n ∈ N.
Calculate the following probabilities (and provide appropriate justifications for your
reasoning):
(b):
(d):
(f):
P[X = Y ]
P[Y > X]
P[X divides Y ]
P[min(X, Y ) ≤ k], where k ∈ N
P[X ≥ kY ], where k ∈ N
(c):
(e):
Hint: The correct final results are among the following:
1
,
3
1−
1
,
4k
1
2k+1
−1
,
∞
X
k=1
1
.
2k+1 − 1
Continues on next page. . .
3.3 Independence of three sigma-algebras. Let (Ω, F, P) be a probability space and let
G1 , G2 , G3 ⊂ F be sigma-algebras on Ω. Assume that Gk is generated by a π-system
Ik which contains Ω.
(a) Show that G1 , G2 , G3 are independent if and only if
P I1 ∩ I2 ∩ I3 = P[I1 ] P[I2 ] P[I3 ]
for all I1 ∈ I1 , I2 ∈ I2 , I3 ∈ I3 .
(b) Why did we require that Ik contains Ω?
3.4 Infinite-horizon events. Let X1 , X2 , X3 , . . . be a sequence of random numbers (i.e.,
R-valued random variables) defined on a probability space (Ω, F,
TP). Investigate
which of the following events belong to the tail sigma-algebra T∞ = n≥1 σ(Xn+1 , Xn+2 , . . . ).
n
o
(a):
ω ∈ Ω lim Xn (ω) exists
n→∞
∞
X
o
n
Xn (ω) converges
ω∈Ω
(b):
n=1
(c):
(d):
(e):
∞
X
n
o
ω∈Ω
Xn (ω) ≤ 42
n=1
n
o
ω ∈ Ω sup Xn (ω) < ∞
n∈N
n
o
ω ∈ Ω ∀ ` ∈ N ∃n ∈ N such that Xn (ω) = Xn+1 (ω) = · · · = Xn+` (ω)
= {there exists arbitrarily long repetitions in the sequence X1 , X2 , . . .}
3.5 Monkey typing Aleksis Kivi. A monkey has a keyboard of 50 keys (no shift, no
caps lock, no control keys), and keeps hitting the keys randomly one at a time.
Assume that the (infinite) string (X1 , X2 , . . . ) produced by the monkey is such that
the
X1 , X2 , . . . are independent and uniformly distributed, so that
random
symbols
1
P Xn = x = 50
for each symbol x in the alphabet.
The novel Seitsemän veljestä contains N = 635864 symbols (upper and lower case
letters identified).
(a) Compute the probability that the monkey types the novel Seitsemän veljestä
right from the start (in the first N symbols typed by the monkey).
(b) Compute the probability that the monkey ever types Seitsemän veljestä (among
any N consecutive symbols).
(c) Compute the probability that the monkey types infinitely many copies of Seitsemän veljestä (each copy consisting of some N consecutive symbols).
Hint: Borel–Cantelli lemma.