Page 58 - MATLAB Recipes for Earth Sciences

3.4 Theoretical Distributions 49

Probability Density Cumulative Distribution
Function Function
1 1
Φ 1 =1, Φ 2 =5
0.8 Φ 1 =10, Φ 2 =10 0.8

0.6 0.6
f(x) Φ 1 =1, Φ 2 =5 F(x) Φ 1 =10, Φ 2 =10
0.4 0.4

0.2 0.2

0 0
0 1 2 3 4 0 1 2 3 4
x x
a b
Fig. 3.10 a Probability density function f(x) and b standardized (F(x) =1) cumulative
max
distribution function F(x) of a Fisher¶s F distribution with different values for Φ and Φ .
1 2

plex probability density function:

where x>0 and Γ is again the Gamma function. The two parameters Φ and
1
Φ are the degrees of freedom.
2

χ or Chi-Squared Distribution
2
The χ distribution was introduced by Friedrich Helmert (1876) and Karl
2
Pearson (1900). It is not used for fitting a distribution, but has important ap-

2
plications in statistical hypothesis testing, namely the χ –test (Chapter 3.9).
2
The probability density function of the χ distribution is

where x>0, otherwise f(x)=0. Again, Φ is the degrees of freedom (Fig. 3.11).