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Advanced Probability EPFL - Fall Semester 2012-
Final Exam
SURNAME: ............................................... FIRST NAME: ...........................................................
Exam duration: 3 hours
Allowed material: four handwritten single-sided A4 pages
Exercise 1. Let X be a random variable with pdf
p X
(x) =
π
1 + x
2
, if x ≥ 0 ,
0 , if x < 0.
a) Bonus quiz. Which of the following statements are correct? (no justifications required)
a1) X is a continuous random variable; a4) P({X ≥ 1 }) =
1
2
a2)
E(X) ≤ E(
X); a5) E(X) = 1.
a3) P({X ≥ t}) ≤
E(
√
X) √
t
for all t > 0;
Let also Z = log(X).
b) Compute the pdf of Z, as well as E(Z) and Var(Z).
Hint: The following integral might be useful for this question:
R
dt
t
2
cosh(t)
π
3
, where we recall that cosh(t) =
e
t
−t
Let now (X n
, n ≥ 1) be i.i.d. random variables, all with the same distribution as X, and let
Yn = X 1 · X 2 · · · Xn, n ≥ 1.
c) Does the sequence (Y
1 /n
n
, n ≥ 1) converge to a limiting random variable Y 0
as n → ∞? In which
sense? If it exists, what is this random variable Y 0
? Justify your answers.
d) Does the sequence (Y
1 /
√
n
n
, n ≥ 1) converge to a limiting random variable Y as n → ∞? In
which sense? If Y exists, what is its distribution? Justify your answers.
e) Bonus. Show that for t > 2, there exists c > 0 such that
P({Yn ≥ t
n
}) ≤ exp(−nc), for n sufficiently large.
Hint: The following integral might be useful for this question:
∞
0
dx
x
1 + x
2
π
please turn the page %
Exercise 2. Let (X n
, n ≥ 1) be a sequence of i.i.d. random variables such that P({X 1
P({X
1
= 0}) = 1/2. Let also F 0
= {∅, Ω} and F n
= σ(X 1
,... , X
n
) for n ≥ 1. Finally, let
Y = (Y
n
, n ∈ N) be the process defined as
Y 0 = 0, Yn =
n ∑
j=
X
j
j
, n ≥ 1.
a) Is the process Y a submartingale, supermartingale or martingale with respect to the filtration
(F
n
, n ∈ N)? Justify your answer.
b) Compute E(Yn) and Var(Yn) for all n ≥ 1.
c) Is the process Y confined to some interval? In case the answer is yes, what is this interval? In
case the answer is no, explain why.
d) Does there exist a deterministic process A such that Y − A is a martingale? In case the answer
is yes, compute this process recursively; in case the answer is no, explain why.
e) Does there exist a random variable Y∞ such that Yn →
n→∞
Y∞ almost surely? Justify your
answer.
f ) If it exists, what is the distribution of the random variable Y ∞
? (no formal justification required
here)
Exercise 3. Let Y = (Y n
, n ∈ N) be the process defined recursively as
Y
0
= 1, Y
n+
3 Y
n
, with probability 1/ 2 ,
Y
n
, with probability 1/ 2.
a) Is the process Y a submartingale, supermartingale or martingale with respect to its natural
filtration (Fn, n ∈ N)? Justify your answer.
b) Compute E(Yn) and Var(Yn) recursively, for all n ≥ 1.
c) Is the process Y confined to some interval? In case the answer is yes, what is this interval? In
case the answer is no, explain why.
d) Does there exist a random variable Y ∞
such that Y n
n→∞
Y
∞
almost surely? Justify your
answer.
e) If it exists, what is the random variable Y ∞
Hint: In order to answer this question rigorously, consider the process Z defined as Z n
= log(Y n
f ) If Y∞ exists, does it also hold that Yn = E(Y∞|Fn)? Justify your answer.
