Information Theory and Principal Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 83

0.8 0.8 1.5

0.6 0.6

1

0.4 0.4

0.5

0.2 0.2

0 0 0

2 0 2 2 0 2 1 0 1

Figure 3.4
Random variables with different kurtosis. In each picture a Gaussian (kurt = 0)
with zero mean and unit variance is plotted in dashed lines. The left figure shows a

Laplacian distribution with λ = √ 2. In the middle figure a uniform density in

[ − √ 3 , √ 3] is shown. It has zero mean and kurtosis − 1 . 2. The right picture shows
the sub-Gaussian random variable X := π cos( Y )with Y uniform in [ −π, π ]. Its
kurtosis is − 2 8 , [105]. Figures courtesy of Dr. Christoph Bauer [19].

tokurtic. If kurt( X ) < 0, X is called sub-Gaussian or platykurtic. If
kurt( X )=0, X is said to be mesokurtic .

By lemma 3.7, Laplacians are superGaussian, and uniform densities
are sub-Gaussian densities. In practice, superGaussian variables are
often pictured as having sharper peaks and longer tails than Gaussians,
whereas sub-Gaussians tend to be flatter or multimodal, as those two
examples confirm. See figure 3.4 for these and more examples.

Sampling

Above, we spoke about only random functions. In actual experiments
those are not known, but some samples (i.e. some values) of the random
function are known. Sampling is defined in this section.

Definition 3.16: Given a finite independent sequence ( X i ) i=1,...n of
random functions on a probability space (Ω , A ,P ) with the same distri-
bution function F and an element ω

Ω. Then the n elements X i ( ω ),
i =1 ,...,n are called i.i.d. samples of the distribution F .

∈