Probabilities are often stated in terms of cumulative distribution functions (CDFs). The CDF
F
x
(
a
)
of a random
variable
x
is the probability that the random variable
x
takes on a value less than or equal to
a
, viz.
F
x
(
a
)
=
P
(
x
≤
a
)
F
x
(
a
)
is a nondecreasing function of its argument that takes on values between 0 and 1. Another important func
tion is the probability density function (PDF)
f
x
(
x
)
of the random variable
x
, which is related to the CDF through
integration/differentiation:
F
x
(
a
)
=
P
(
x
≤
a
)
=
integraldisplay
a
−∞
f
x
(
x
)
dx
dF
x
(
x
)
dx
=
f
x
(
x
)
The PDF is the probability of observing the variable
x
in the interval
(
x,x
+
dx
)
. Since a random variable always has
some value, we must have
integraldisplay
∞
−∞
f
x
(
x
)
dx
=
1
Note that, for continuous random variables, a PDF of 0 (1) for some outcome does not mean that that outcome is
prohibited (guaranteed). Either the PDF or the CDF completely determine the behavior of the random variable and are
therefore equivalent.
Random variables that are normal or Gaussian occur frequently in many contexts and have the PDF:
f
N
(
x
)
=
1
σ
√
2
π
e
−
1
2
(
x
−
μ
)
2
/σ
2
which is parametrized by
μ
and
σ
. Normal distributions are often denoted by the shorthand
N
(
μ,σ
)
. Another common
distribution is the Rayleigh distribution, given by
f
r
(
r
)
=
braceleftbigg
r
σ
2
e
−
r
2
/
2
σ
2
,r
≥
0
0
,r<
0
which is parametrized just by
σ
.
It can be shown that a random variable defined as the length of a 2d Cartesian
vector with component lengths given by Gaussian random variables has this distribution. In terms of radar signals, the
envelope of quadrature Gaussian noise is Rayleigh distributed. Still another important random variable called
χ
2
has
the PDF
f
χ
2
=
1
2
ν/
2
Γ(
ν/
2)
x
1
2
ν
−
1
e
−
x/
2
where
Γ
is the Gamma function,
Γ(
x
) =
integraltext
∞
0
ξ
x
−
1
e
−
xi
dξ
, and where the parameter
ν
is referred to as the number of
degrees of freedom. It can be shown that if the random variable
x
i
is
N
(0
,
1)
, then the random variable
z
=
n
summationdisplay
i
=1
x
2
i
is
χ
2
with
n
degrees of freedom. If
x
i
refers to voltage samples, for example, then
z
refers to power estimates based
on
n
voltage samples or realizations.
20
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Mean and variance
It is often not the PDF that is of interest but rather just its moments. The expectation or mean of a random variable
x
,
denoted
E
(
x
)
or
μ
x
, is given by
E
(
x
)
=
integraldisplay
∞
−∞
xf
x
(
x
)
dx
For example, it can readily be shown that the expected value for a normal distribution
N
(
μ,σ
)
is
E
(
x
) =
μ
. The
variance of a random variable, denoted
Var
(
x
)
or
σ
2
x
, is given by
Var
(
x
)
=
E
[(
x
−
μ
x
)
2
]
=
E
(
x
2
)
−
μ
2
x
=
integraldisplay
∞
−∞
(
x
−
μ
x
)
2
f
x
(
x
)
dx
The standard deviation of
x
meanwhile, which is denoted
σ
x
, is the square root of the variance. In the case of the same
normal distribution
N
(
μ,σ
)
, the variance is given by
σ
2
, and the standard deviation by
σ
.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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