Stochastic Signals and Systems
Multiple Random Variables
Virginia Tech
Fall 2008
Jointly Gaussian RVs
Let
X
and
Y
be independent RVs with the distributions:
•
X
is
N
(
0
,σ
2
)
.
•
Y
takes the value 1 with probability
p
and

1 with
probability
q
=
1

p
.
Deﬁne
W
=
XY
.
(a) Show that
W
is Gaussian.
(b) Find the joint distribution of
X
and
W
. Are these variables
jointly
Gaussian?
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View Full DocumentCentral Limit Theorem
Let
S
n
be the sum of
n
independent, identically distributed
random variables with ﬁnite mean
E
[
X
] =
μ
and ﬁnite variance
σ
2
, and let
Z
n
be the zeromean, unitvariance random variable
deﬁned by
Z
n
=
S
n

n
μ
σ
√
n
then
lim
n
→∞
P
[
Z
n
≤
z
] =
1
√
2
π
Z
z
∞
e

x
2
/
2
dx
.
In other words,
F
Z
n
(
z
)
converges to the standard normal cdf.
The ubiquity of Gaussian rvs in science and engineering can be
attributed in part to the fact that errors in measurements and noise in
systems are often the sums of great many small, random integrants,
and by virtue of the central limit theorem, the distributions of such
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 Fall '08
 DASILVER
 Central Limit Theorem, Normal Distribution, Variance, Probability theory

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