Mathematics 136,
Midterm 2008
Write your name and sign the Honor code in the blue books provided.
This is a closed material exam, but you may use three pages (2 sides each) of notes.
You have 90 minutes
to solve all
three
questions, each worth points as marked (maximum of 50). Complete reasoning is required
for full credit. You may cite lecture notes and homework sets, as needed, stating precisely the result you use,
why and how it applies.
1.
(4x4) Provide the complete, accurate and rigorous deFnition of
four
of the following concepts (it is not
enough to merely cite their location in text).
a) A measurable space (Ω
,
F
), the structure of
F
, and a probability measure
P
on it.
ANS:
F
is a
σ
Feld of subsets of Ω and detail DeFnitions 1.1.1 and 1.1.2.
b) The law of a random variable
X
, its distribution function and the convergence in
law
of random variables
X
n
to a random variable
X
.
ANS:
Detail DeFnitions 1.4.1, 1.4.5 and 1.4.10.
c) The conditional expectation of
X
∈
L
2
(Ω
,
F
,
P
) given a random variable
Y
on (Ω
,
F
).
ANS:
Detail DeFnition 2.1.3.
d) A stochastic process
{
X
t
, t
∈
IR
}
, its Fnite dimensional distributions and the conditions which make
them
consistent
.
ANS:
Detail DeFnitions 3.1.1, 3.1.5 and 3.1.13.
e) A Gaussian stochastic process
{
X
t
}
and its autocovariance and mean functions
ρ
(
t, s
) and
μ
(
t
).
ANS:
Detail DeFnition 3.2.17 and ideally also DeFnition 3.2.8 for a Gaussian stochastic process
X
t
.
Then,
μ
(
t
) =
E
X
t
and
ρ
(
t, s
) =
E
[(
X
t

μ
(
t
)(
X
s

μ
(
s
))].
2.
(2+3+2+3+3+3)
a) Suppose
N
is a random variable that is uniformly distributed in the Fnite set
{
1
,
2
, . . . , m
}
for some
nonrandom Fnite integer
m
.
±ixing 0
< p <
1 let
{
ξ
k
}
be independent and identically distributed
random variables on the same probability space (Ω
,
F
,
P
), that are independent of
N
and such that
p
=
P
(
ξ
k
= 1) = 1

P
(
ξ
k
=

1). Explain why for any nonrandom constant
r
,
Y
(
ω
) =
N
(
ω
)
X
k
=1
(1 +
rξ
k
(
ω
))
,
can be written as
Y
=
∑
m
k
=1
(1 +
rξ
k
)
I
{
N
≥
k
}
and why it is in
L
1
(Ω
,
F
,
P
).
ANS:
Applying the identity
g
(
n, a
1
, . . . , a
m
) =
∑
n
k
=1
a
k
=
∑
m
k
=1
a
k
1
n
≥
k
which is valid for any
a
k
∈
IR
1