come. The focus of elementary statistics (Section 3.4.2) is to describe the underlying
probabilistic parameters by functions on the experimental sample, the sample statis-
tics, and to provide tools for visualization of the data. Statistical test theory (Section
3.4.3) provides a framework for judging the significance of statements (hypotheses)
with respect to the data. Linear models (Section 3.4.4) are one of the most prominent
tools for analyzing complex experimental procedures.
3.4.1
Basic Concepts of Probability Theory
The quantification and characterization of uncertainty are formally described by the
concept of a probability space for a random experiment. A random experiment is an
experiment that consists of a set of possible outcomes with a quantification of the
possibility of such an outcome. For example, when a coin is tossed, one cannot deter-
ministically predict the outcome of “heads” or “tails” but rather assigns a probability
that either of the outcomes will occur. Intuitively, one will assign a probability of 0.5
if the coin is fair (i.e., both outcomes are equally likely). Random experiments are
described by a set of probabilistic axioms.
A probability space is a triplet (O, A, P) where O is a nonempty set, A is a s-alge-
bra of subsets of O, and P is a probability measure on A.As-algebra is a family of
subsets of O that (1) contains O itself, (2) contains for every element B B A the com-
plementary element B
c
B A, and (3) contains for every series of elements B
1
, B
2
,…,
B A their union, i.e.,
S
1
i1
B
i
B A. The pair (O, A) is called a measurable space. An
element of A is called an event. If O is discrete, i.e., it has at most countable many
elements, then a natural choice of A would be the power set of O, P(O), i.e., the set
of all subsets of O.
A probability measure P:A?[0,1] is a real-valued function that has the properties
P(B) 6 0 for all B B A and P(O)=1
and
P(
S
1
i1
B
i
)=
P
1
i1
P(B
i
) for all series of disjoint sets B
1
, B
2
,…,B A (s-additivity). (3-59)
78
3 Mathematics in a Nutshell
Fig. 3.5 Bifurcation diagram of the logistic equa-
tion. For increasing parameter r, the number of
steady-states increments are shown. At the points
r =1,r = 3, and r = 3.3, stability changes from
stable (solid lines) to unstable (dashed lines) oc-
cur. Only the constant solution and the solution of
periods 1 and 2 are shown.