Image Analysis, Classification and Change Detection in Remote Sensing, 3rd Edition

40 Image Statistics

As a simple example, consider a uniformly distributed random variable Z

with density function

p(z) =

n

1 if 0 ≤ z ≤ 1

0 otherwise.

We calculate the moments to be

hZi =

Z

1

0

z · 1 dz = 1/2

var(Z) =

Z

1

0

(x − 1/2)

2

·1 dz = 1/12.

Since the populations we are dealing with in the case of actual measurements

are infinite, it is clear that mea n and variance can, in reality, never be known

exactly. As we shall discuss Section 2.3, they must be estimated from the

available data.

Two very useful identities follow from the definition of variance (Exercise 1):

var(Z) = hZ

2

i − hZi

2

var(a

0

+ a

1

Z) = a

2

1

var(Z).

(2.13)

2.1.3 Random vectors

The idea of a distribution ...