Graphics Programs Reference
In-Depth Information
Probability Density
Function
Cumulative Distribution
Function
1
1
Φ
1
=1, Φ
2
=5
0.8
0.8
Φ
1
=10, Φ
2
=10
0.6
0.6
Φ
1
=10, Φ
2
=10
Φ
1
=1, Φ
2
=5
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
)
max
=1) cumulative
distribution function
F
(
x
) of a Fisher¶s F distribution with different values for Φ
1
and Φ
2
.
plex probability density function:
where
x
>0 and
Γ
is again the Gamma function. The two parameters
Φ
1
and
Φ
2
are the degrees of freedom.
χ
2
or Chi-Squared Distribution
The
distribution was introduced by Friedrich Helmert (1876) and Karl
Pearson (1900). It is not used for fi tting a distribution, but has important ap-
plications in statistical hypothesis testing, namely the
χ
2
χ
2
-test (Chapter 3.9).
The probability density function of the
χ
2
distribution is
where
x
>0, otherwise
f
(
x
)=0. Again,
Φ
is the degrees of freedom (Fig. 3.11).
