Stochastic Signals and Systems
Multiple Random Variables
Virginia Tech
Fall 2008
Multiple Random Variables
•
We now develop techniques for calculating the probabilities
of events that involve the
joint
behavior of two or more rvs.
•
We are also interested in determining when a set of rvs are
independent, as well as in quantifying their degree of
“
correlation
” when they are not independent.
•
Example: Given two random variables
X
and
Y
we wish to
determine their joint statistics, that is, the probability that
the point
(
X
,
Y
)
is in a speciﬁed region
D
in the
xy
plane.
•
In particular, the probability of the event
{
X
≤
x
} ∩ {
Y
≤
y
}
=
{
X
≤
x
,
Y
≤
y
}
cannot be expressed in terms of
F
X
(
x
)
and
F
Y
(
y
)
.
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View Full DocumentJoint Cumulative Distribution Function
For any pair of numbers
x
and
y
, the event
(
X
≤
x
,
Y
≤
y
)
is
represented by a quadrant of the XYplane having its vertex at
the point
(
x
,
y
)
and opening to the lower left, as shown below.
The probability of this event is denoted by
F
X
,
Y
(
x
,
y
) =
P
[
X
≤
x
,
Y
≤
y
]
and is called the
joint cumulative distribution function
of
X
and
Y
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 Fall '08
 DASILVER
 Probability theory, probability density function, Cumulative distribution function

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