Graphics Programs Reference
In-Depth Information
Probability Density
Function
Cumulative Distribution
Function
1
1
σ=0.5
σ=1.0
0.8
0.8
σ=0.65
σ=0.65
σ=0.5
0.6
0.6
σ=1.0
0.4
0.4
0.2
0.2
0
0
−6
−4
−2
0
2
4
6
−6
−4
−2
0
2
4
6
x
x
a
b
Fig. 3.8 a
Probability density function
f
(
x
) and
b
standardized (
F
(
x
)
max
=1) cumulative
distribution function
F
(
x
) of a logarithmic normal distribution with mean =0 and with
different values for
σ.
where
x
>0. The distribution can be described by the two parameters mean
µ
and variance
. The formulas for mean and variance, however, are differ-
ent from the ones used for normal distributions. In practice, the values of
x
are logarithmized, the mean and variance are computed using the formulas
for the normal distribution and the empirical distribution is compared with
a normal distribution.
σ
2
Student
·
s t Distribution
The
Student·s t distribution
was fi rst introduced by William Gosset (1876-
1937) who needed a distribution for small samples (Fig. 3.9). W. Gosset was
a Irish Guinness Brewery employee and was not allowed to publish research
results. For that reason he published his t distribution under the pseudonym
Student
(Student, 1908). The probability density function is
where
Γ
is the Gamma function
