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3.4 Theoretical Distributions 47
Probability Density Cumulative Distribution
Function Function
1 1
σ=0.5
0.8 0.8 σ=1.0
σ=0.65 σ=0.65
σ=0.5
0.6 0.6
f(x) σ=1.0 F(x)
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) =1) cumulative
max
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
2
and variance σ . 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.
Student·s t Distribution
The Student·s t distribution was first 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
