Lecture Notes on the Gaussian Distribution
Hairong Qi
The Gaussian distribution is also referred to as the normal distribution or the
bell curve distribution for its bell-shaped density curve. There’s a saying that
within the image processing and computer vision area, you can answer all ques-
tions asked using a Gaussian. The Gaussian distribution is also the most popularly
used distribution model in the field of pattern recognition. So let’s take a closer
look at it.
1 The Definition
The formula for a d-dimensional Gaussian probability distribution is
p(x) =
1
(2π)
d/2
|Σ|
1/2
exp(−
(x − µ)
T
Σ
−1
(x − µ)
2
) (1)
where x is a d-element column vector of variables along each dimension, µ is the
mean vector, calculated by
µ = E[x] =
Z
xp(x)dx
and Σ is the d × d covariance matrix, calculated by
Σ = E[(x − µ)(x − µ
T
] =
Z
(x − µ)(x − µ)
T
p(x)dx
with the following form.
σ
11
σ
12
· · · σ
1d
σ
21
σ
22
· · · σ
2d
· · · · · · · · · · · ·
σ
d1
σ
d2
· · · σ
dd
(2)
1
