Denker
SPRING 2010
416 Stochastic Modeling  Assignment 2
SOLUTIONS
Problem 1:
E
[
X

Y
=
y
] means the expectation of the distribution of
X
when the outcome
y
of
the random variable
Y
is known.
First we calculate
f
X

Y
(
x

y
).
f
Y
(
y
) =
integraldisplay
y

y
y
2

x
2
8
e

y
dx
=
1
6
y
3
e

y
.
f
X

Y
(
x

y
) =
y
2

x
2
8
e

y
1
6
y
3
e

y
=
3
4
(
1
y

x
2
y
3
)
.
E
[
X

Y
=
y
] =
integraldisplay
y

y
x
3
4
(
1
y

x
2
y
3
)
dx
= 0
,
since the integrand is point symmetric around 0.
Problem 2:
Let
N
A
(resp.
N
B
) denote the number of plays needed to win two in a
row when beginning to play with A (resp. B). Let
X
1
, X
2
, ...
denote the sequence of
independent random variables of the game when beginning with opponent A.
Define a random variable
Y
by
Y
= 1, if
X
1
= 0,
Y
= 2, if
X
1
= 1
, X
2
= 1, and
Y
= 3,
if
X
1
= 1
, X
2
= 0. Then
P
(
Y
= 1) = 1

p
A
,
P
(
Y
= 2) =
p
A
p
B
and
P
(
Y
= 3) =
p
A
(1

p
B
)
and
E
[
N
A
] =
E
[
N
A

Y
= 1]
P
(
Y
= 1) +
E
[
N
A

Y
= 2]
P
(
Y
= 2) +
E
[
N
A

Y
= 3]
P
(
Y
= 3)
.
If
Y
= 1, you plaed once with no win, so you start the same game playing B first. So
the distribution of
N
A
given
Y
= 1 equals that of 1 +
N
B
, so
E
[
N
A

Y
= 1] = 1 +
E
[
N
B
]
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Prof.Denker
 Stochastic Modeling, Probability theory, NA Y

Click to edit the document details