Modeling with Neural Networks: Principles and Model Design Methodology - Neural Networks: Methodology and Applications - page 88

0,4

0,35

0,3

0,25

0,2

0,15

0,1

0,05

0

-5

0

5

y

Fig. 2.1. Normal distribution

2.2.1.1 Examples of Probability Distributions

Uniform Distribution

A random variable Y has a uniform distribution if its density probability is
p Y ( y )=1 / ( b − a )onagiveninterval[ a,b ], and is zero elsewhere.

Gaussian Distribution
The Gaussian distribution p Y ( y )=1 / ( √ 2 πσ 2 ) exp( µ ) 2 ) / (2 σ 2 )) is very

useful. µ is the mean of the Gaussian and σ ( > 0) is its standard deviation.
Figure 2.1 shows a normal distribution , with µ =0and σ =1.

− (( y −

Other Distributions

The Pearson (or χ 2 ) distribution, the Student distribution and the Fisher
distribution are defined in the additional material at the end of the chapter.

2.2.1.2 Joint Distributions

Denoting by p X,Y ( x,y ) the joint density of two random variables, the proba-
bility that a realization of X lie between x and x +d x and that a realization
of Y lie between y and y +d y is p X,Y ( x,y )d x d y .

Independent Random Variables

If two random variables X and Y are independent, one has:

p X,Y ( x,y )= p X ( x ) p Y ( y ) .