It is important to distinguish between the mean and variance of a random variable and the sample mean and
sample variance of the realizations of a random variable. The latter are experimental quantities constructed from a
finite number of realizations of the random variable
x
. The latter will depart from the former due to the finite sample
size and perhaps because of estimator biases. The deviations constitute experimental errors, and understanding them
is a substantial part of the radar art.
Joint probability distributions
Two random variables
x
and
y
may be described by a joint probability density function (JPD)
f
(
x,y
)
such that
P
(
x
≤
a
and
y
≤
b
)
=
integraldisplay
a
−∞
integraldisplay
b
−∞
f
(
x,y
)
dydy
In the event the two random variables are independent, then
f
(
x,y
) =
f
x
(
x
)
f
y
(
y
)
. More generally, we can define the
covariance of
x
and
y
by
Cov
(
x,y
)
=
E
[(
x
−
μ
x
)(
y
−
μ
y
)]
=
E
(
xy
)
−
μ
x
μ
y
If
x
and
y
are independent, their covariance will be 0. It is possible for
x
and
y
to be dependent and still have zero
covariance. In either case, we say that the random variables are uncorrelated.
The Correlation function is defined by
ρ
(
x,y
)
=
Cov(
x,y
)
radicalbig
Var(
x
)Var(
y
)
The variance, covariance, and correlation function conform to the following properties (where
a
is a scalar and
x
and
y
are random variables):
1. Var(
x
)
≥
0
2. Var(
x
+
a
) = Var(
x
)
3. Var(
ax
) =
a
2
Var(
x
)
4. Var(
x
+
y
) = Var(
x
)+2Cov(
x,y
)+Var(
y
)
5.
ρ
(
x,y
)
≤
1
These ideas apply not only to different random variables but also to realizations of the same random variable at different
times. In this case, we write of autocovariances and autocorrelations evaluated at some time lag (see below).
21
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Conditional probability
It is often necessary to determine the probability of an event given that some other event has already occurred. The
conditional probability of event
A
given event
B
having occurred is
P
(
A

B
)
=
P
(
A
∩
B
)
/P
(
B
)
Since
P
(
A
∩
B
)
=
P
(
B
∩
A
)
, the equation could equally well be written with the order of
A
and
B
reversed. This
leads immediately to Bayes’ theorem, which gives a prescription for reversing the order of conditioning:
P
(
B

A
)
=
P
(
A

B
)
P
(
B
)
P
(
A
)
Given conditional probabilities, one can construct conditional PDFs
f
x

y
(
x
)
, conditional CDFs
F
x

y
(
a
)
, and also their
moments.
Multivariate normal distribution
Consider now the joint probability distribution for a column vector
x
with
n
random variable components. If the
components have a multivariate normal (MVN) distribution, their joint probability density function is
f
(
x
)
=
1
(2
π
)
n/
2
1
radicalbig
det(
C
)
e
−
1
2
(
x
−
u
)
t
C
−
1
(
x
−
u
)
(1.14)
where
u
is a column vector composed of the expectations of the random variables
x
and
C
is the covariance matrix
for the random variables. This definition assumes that the covariance matrix is nonsingular.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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