Databases Reference
In-Depth Information
The mean of the uniform distribution can be obtained as
b
+
1
b
a
μ
X
=
=
x
a
dx
b
−
2
a
Similarly, the variance of the uniform distribution can be obtained as
2
=
(
b
−
a
)
2
X
σ
12
Details are left as an exercise.
A.5.2 Gaussian Distribution
This is the distribution of choice in terms of mathematical tractability. Because of its form, it
is especially useful with the squared error distortion measure. The probability density function
for a random variable with a Gaussian distribution is
2
1
−
(
x
−
μ)
f
X
(
x
)
=
√
2
2
exp
(A.16)
2
2
σ
πσ
2
.
where the mean of the distribution is
μ
and the variance is
σ
A.5.3 Laplacian Distribution
Many sources that we will deal with will have probability density functions that are quite
peaked at zero. For example, speech consists mainly of silence; therefore, samples of speech
will be zero or close to zero with high probability. Image pixels themselves do not have
any attraction to small values. However, there is a high degree of correlation among pixels.
Therefore, a large number of the pixel-to-pixel differences will have values close to zero. In
these situations, a Gaussian distribution is not a very close match to the data. A closer match
is the Laplacian distribution, which has a
pdf
that is peaked at zero. The density function for
a zero mean random variable with Laplacian distribution and variance
2
is
σ
2
exp
−
√
2
1
√
2
|
x
|
f
X
(
x
)
=
(A.17)
σ
σ
A.5.4 Gamma Distribution
A distribution with a
pdf
that is even more peaked, though considerably less tractable than the
Laplacian distribution, is the gamma distribution. The density function for a gamma-distributed
random variable with zero mean and variance
2
is given by
σ
√
3
exp
−
√
3
4
|
x
|
f
X
(
x
)
=
√
8
(A.18)
2
σ
πσ
|
x
|
