ECSE305, Winter 2009
Probability and Random Signals I
Assignment #8
Posted:
Thursday, March 19, 2009.
Due:
Thursday, March 26, 2009, 2h30pm.
Important notes:
•
Assignments without this cover page will be discarded.
Student #1:
Name:
ID:
Student #2:
Name:
ID:
Question
Marks
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Total
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1. The joint PDF of random variables
X
and
Y
is given by
f
(
x, y
) =
braceleftBigg
cx
k
y
l
,
0
≤
x
≤
1
,
0
≤
y
≤
1
0
,
otherwise
where
k
and
l
are positive integers.
(a) Find an expression for the constant
c
in terms of
k
and
l
(b) Find a necessary condition on the values of
k
and
l
such that
P
(
Y > X
)
>
1
/
2.
(c) Find the marginal PDFs of
X
and
Y
.
2. Using an appropriate software package (e.g.
Matlab), sketch the 3D
graph of the joint normal PDF
N
(
μ
X
, μ
Y
, ρ, σ
2
X
, σ
2
Y
) for the following
parameter values (note: handdrawing will not be accepted):
(a)
μ
X
=
μ
Y
= 0,
σ
X
=
σ
Y
= 1 and
ρ
= 0,;
(b)
μ
X
= 1
, μ
Y
= 1
/
2,
σ
X
=
σ
Y
= 1 and
ρ
= 0,;
(c)
μ
X
=
μ
Y
= 0,
σ
X
= 1
, σ
Y
= 2 and
ρ
= 0,;
(d)
μ
X
=
μ
Y
= 0,
σ
X
=
σ
Y
= 1 and
ρ
= 0
.
5.
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 Spring '09
 Champagne
 Normal Distribution, Variance, Probability theory, probability density function, Cumulative distribution function, joint probability density

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