Graphics Programs Reference
In-Depth Information
∂
x
∂
x
∂
u
∂
v
J
=
(13.75)
∂
y
∂
y
∂
u
∂
v
where the determinant of the matrix of derivatives
J
is called the Jacobian.
The characteristic function for the random variable
X
is defined as
∞
∫
C
X
()
Ee
j
ω
X
f
X
()
e
j
ω
x
=
[
]
=
d
(13.76)
∞
The characteristic function can be used to compute the
pdf
for a sum of inde-
pendent random variables. More precisely, let the random variable
Y
be equal
to
Y
=
+++
X
2
…
X
N
(13.77)
1
where
{
X
i
;
i
=
1 …
N
,
}
is a set of independent random variables. It can be
shown that
C
Y
()
C
X
1
()
C
X
2
()…
C
X
N
()
=
(13.78)
and the
pdf
f
Y
()
is computed as the inverse Fourier transform of
C
Y
()
(with
the sign of
y
reversed),
∞
∫
1
2π
j
ω
y
------
f
Y
()
=
C
Y
()
e
d
(13.79)
∞
The characteristic function may also be used to compute the
nth
moment for
the random variable
X
as
n
d
EX
[]
j
(
n
=
C
X
()
ω 0
(13.80)
ω
n
d
=
13.7. Multivariate Gaussian Distribution
Consider a joint probability for
m
random variables,
X
1
,
X
2
…
X
m
,
,
. These
variables can be represented as components of an
m
×
1
random column vec-
tor,
X
. More precisely,
Search WWH ::
Custom Search