Homework 4
On-campus students:
Submit answers to all problems not labeled SS by the end of class on
Friday, October 21st.
Off-campus students:
Submit answers to all problems not labeled SS by Friday, October 28th.
J
OINT
D
ENSITIES AND
C
ONDITIONAL
D
ENSITIES
I
NVOLVING
M
ULTIPLE
R
ANDOM
V
ARIABLES
SS-1.
X
and
Y
are random variables with joint density function
f
XY
(
x
,
y
) =
ce
-
(
x
2
+
y
2
)
/
2
,
x
2
+
y
2
≤
1
0
,
otherwise
,
where
c
is an appropriately chosen constant.
(a) Find the value of
c
.
Answer:
c
=
2
π
(
1
-
e
-
0
.
5
)
-
1
(b) Find the marginal density of
Y
.
Answer:
f
Y
(
y
) =
ce
-
y
2
/
2
√
2
π
1
-
2
Q
1
-
y
2
,
-
1
≤
y
<
1
0
,
otherwise
(c) Find the conditional density of
X
given
Y
.
Answer:
f
X
|
Y
(
x
|
y
) =
e
-
(
x
2
+
y
2
)
/
2
e
-
y
2
/
2
√
2
π
1
-
2
Q
√
1
-
y
2
,
-
1
-
y
2
≤
x
<
1
-
y
2
and
-
1
≤
y
≤
1
0
,
otherwise
(d) Sketch the shape of the region where
|
XY
| ≤
√
3
/
4
≈
0
.
433. Solve for the important
points that define the region.
1.
X
and
Y
are random variables with joint density
f
XY
(
x
,
y
) =
8
xy
,
0
≤
x
≤
y
≤
1
0
,
otherwise
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(a) Find the marginal density for
Y
.
(b) Find the conditional density for
X
given
Y
(c) Are
X
and
Y
statistically independent? Justify your answer.
(d) Find the joint distribution function of
X
and
Y
.
2. Write a MATLAB (or Octave) program to generate 100,000 random variables with the joint
density given in problem 1.
(Hint: I have not explicitly taught you how to generate random
variables that are not independent! Use your knowledge of generating random variables
and your knowledge of conditional probability, distribution functions, etc. to figure it out.)
Plot a two-dimensional histogram (see MATLAB command hist3) of the generated values
and compare with a surface plot of the joint density. As a reminder, you may discuss ideas
about how to solve this problem with your classmates or myself, but you must do the coding
yourself.
SS-2. Let
X
,
Y
, and
Z
denote independent random variables. Find the following probabilities in
terms of
F
X
(
x
)
,
F
Y
(
y
)
, and
F
Z
(
z
)
.
(a)
P
|
X
|
<
5
,
Y
>
2
,
Z
2
≥
2
Answer:
[
F
X
(
5
-
)
-
F
X
(
-
5
)][
1
-
F
Y
(
2
)][
1
-
F
Z
(
√
2
-
)+
F
Z
(
-
√
2
)]
(b)
P
[
X
>
5
,
Y
<
0
,
Z
=
1
]
Answer:
[
1
-
F
X
(
5
)]
F
Y
(
0
-
)[
F
Z
(
1
)
-
F
Z
(
1
-
)]
(c)
P
[
min
(
X
,
Y
,
Z
)
>
2
]
Answer:
[
1
-
F
X
(
2
)][
1
-
F
Y
(
2
)][
1
-
F
Z
(
2
)]
(d)
P
[
max
(
X
,
Y
,
Z
)
<
6
]
Answer:
F
X
(
6
-
)
F
Y
(
6
-
)
F
Z
(
6
-
)
F
UNCTIONS OF
S
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- Spring '11
- Wang
- Probability theory, Woods, probability density function
-
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