Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 400

6.2.1 Basic Notation and Concepts

We summarize some fundamental concepts and notation that are useful in the
remainder of this section. For a complete and rigorous introduction and explanation
of these concepts, we refer, e.g., to [ 55 ].

Random Variables

For a random variable w, i.e., a variable whose value depends on a random
experiment !, we introduce the distribution function

F W . w / P.! W w.!/ w /

where the notation on the right-hand side represents the probability that the
realization of w associated with ! is w . Elementary properties of probability
imply that lim w !1 F W D 0 and lim w !C1 F W D 1 and that the function is non-
decreasing. The corresponding p.d.f. is defined as

dF W
d w :
For the properties of distribution, p W . w / 0 and Z
R

p W . w /

p W d w D 1.

The Gaussian p.d.f. for instance reads

p 2 exp

:

. w / 2
2 2

1

g W . w / D (6.3)

A p.d.f. can be characterized by its moments . In particular, we define the
expectation operator E ./ as

Z

E .w/ w p W . w /d w ;

R

that associates the random variable with a number called mean . Similarly, we may
consider the moments and the central moments of order m defined, respectively, as

Z Z

.w m / w m p W . w /d w ; .w m / .w// m p W . w /d w :

E E

. w E

R R

The central moment of order 2 is called variance . For the Gaussian p.d.f. g W ,the
mean is and variance is 2 .