Mathematics Department, UCLA
T. Richthammer
spring 09, sheet 5
Apr 24, 2009
Homework assignments: Math 170B Probability, Sec. 1
42. Choose a number
n
completely at random from
{
1
,
2
,
3
,
4
}
, then choose a number
k
com
pletely at random from
{
1
, . . . , n
}
. Given the value of
k
, calculate the distribution and
the expected value of the ﬁrst number in all cases
k
= 1
,
2
,
3
,
4.
Answer:
Let
X
denote the ﬁrst number and
Y
the second number. Given that
Y
=
k
, the possible
values of
X
are
k, .
. . ,
4 and we have
p
X
(
n

Y
=
k
) =
P
(
X
=
n,Y
=
k
)
P
(
Y
=
k
)
.
P
(
X
=
n, Y
=
k
) =
1
4
1
n
and
P
(
Y
=
k
) =
P
(
X
=
k, Y
=
k
) +
. . .
+
P
(
X
= 4
, Y
=
k
) =
1
4
(
1
k
+
. . .
+
1
4
). Thus
p
X
(
n

Y
=
k
) =
1
n
1
k
+
...
+
1
4
. Using that formula we get:
Given
Y
= 1,
X
= 1
,
2
,
3
,
4 with probabilities
12
25
,
6
25
,
4
25
,
3
25
and
E
(
X

Y
= 1) =
48
25
= 1
.
92.
Given
Y
= 2,
X
= 2
,
3
,
4 with probabilities
6
13
,
4
13
,
3
13
and
E
(
X

Y
= 2) =
36
13
≈
2
.
77.
Given
Y
= 3,
X
= 3
,
4 with probabilities
4
7
,
3
7
and
E
(
X

Y
= 3) =
24
7
≈
3
.
43.
Given
Y
= 4,
X
= 4 with probability 1 and
E
(
X

Y
= 4) = 4.
43. Let
X, Y
have the joint PDF
f
(
x, y
) = 24
xy
1
{
x
≥
0
,y
≥
0
,x
+
y
≤
1
}
.
(a) Calculate
P
(
Y
≥
1
2

X
= 0),
P
(
Y
≥
1
2

X
=
1
4
),
P
(
Y
≥
1
2

X
=
1
2
).
(b) Calculate
E
(
Y

X
=
1
2
) and
E
(
1
Y

X
=
1
2
).
Answer:
(a) For 0
≤
x
≤
1 we have
f
X
(
x
) =
R
1

x
0
24
xydy
= 12
x
(1

x
)
2
, so
f
Y
(
y

X
=
x
) =
2
y
(1

x
)
2
for 0
≤
y
≤
1

x
. We get
P
(
Y
≥
1
2

X
= 0) =
R
1
1
/
2
2
ydy
=
3
4
,
P
(
Y
≥
1
2

X
=
1
4
) =
R
3
/
4
1
/
2
32
9
ydy
=
5
9
,
P
(
Y
≥
1
2

X
=
1
2
) =
R
1
/
2
1
/
2
8
ydy
= 0.
(b)
f
Y
(
y

X
=
1
2
) = 8
y
1
[0
,
1
/
2]
(
y
) from (a). Thus
E
(
Y

X
=
1
2
) =
R
1
/
2
0
8
y
2
dy
=
1
3
and
E
(
1
Y

X
=
1
2
) =
R
1
/
2
0
8
dy
= 4.
44. Let
X, Y
have the joint PDF
f
(
x, y
) =
e

y
y
1
{
0
<x<y
}
. Calculate
E
(
X
3

Y
=
y
).
Answer:
f
Y
(
y
) =
e

y
, so
f
X
(
x

Y
=
y
) =
1
y
for 0
< x < y
, i.e.
X
is a uniform RV. Using this
density we get
E
(
X
3

Y
=
y
) =
R
y
0
x
3 1
y
=
1
y
1
4
y
4
=
y
3
4
.
45. We choose a point
X
from [0
,
1], and after that we independently choose
X
1
and
X
2
from
[0
, X
] and [
X,
1] respectively. (All choices are made completely at random.)
(a) Calculate the expected distance of
X
1
and
X
2
, given that
X
=
x
.
(b) Explain your result in (a) without any calculation.
(c) Calculate the expected value of
X
, given that
X
1
=
x
1
and
X
2
=
x
2
.
(d) Check your result of (c) in the special case
x
1
=
1
2

c
,
x
2
=
1
2
+
c
for arbitrary
c
∈
[0
,
1
2
]. Which value do you get for
x
1
=
1
4
,
x
2
=
1
2
?