EE 464, Mitra
Spring 2009
Homework 8
Due Friday, March 27, 2009, 9am 464 Box
This problem set investigates tail inequalities, transforms and preliminaries for bivariate random variables.
1. Bivariate Gaussian random variables
(a) Show that the probability density function provided in class for two jointly Gaussian random
variables
[
X, Y
]
T
is equivalent to the following expression:
p
Z
(
z
)
=
1
(2
π
)
n
2
p

Σ

exp

1
2
(
z

m
)
T
Σ

1
(
z

m
)
where
Z
=
[
X, Y
]
T
m
=
[
m
X
, m
Y
]
T
Σ
=
•
σ
2
X
ρσ
X
σ
Y
ρσ
X
σ
Y
σ
2
Y
‚
and
n
=
2
Note that the notation
 · 
denotes taking the determinant.
(b) The Matlab command
x = randn(2,2000)
will generate 2000 samples of a bivariate Gaus
sian random variable
X
= [
X
1
, X
2
]
T
with mean vector and covariance matrix
m
=
•
0
0
‚
Σ =
•
1
0
0
1
‚
We will soon learn that we can convert these two independent, standard Gaussian random
variables into two Gaussians with arbitrary mean and covariance matrix through the following
linear transformation (note that
I
is the twobytwo identity matrix):
X
∼
N
(0
,
I
)
Y
=
AX
+
m
Y
∼
N
(
m
, AA
T
)
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 Spring '06
 Caire
 Normal Distribution, Variance, Gate, Probability theory, probability density function, Cumulative distribution function

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