Probability densities and distributions - Turbulence in the Atmosphere - page 298

Figure 13.1 Upper: A random, stochastic scalar function u(t) in one realization.
The horizontal line represents a value of u ∗ . Lower: The indicator function φ for
that value of u ∗ . Adapted from Lumley and Panofsky ( 1964 ).

13.2 Probability statistics of scalar functions of a single variable
13.2.1 Probability densities and distributions
We'll consider first an ensemble of random (different in each realization), stochas-
tic (irregular), zero-mean, scalar functions of one independent variable, say time t .
Such an ensemble is sometimes called a random process . We'll denote these func-
tions as u(t α) , with α the realization index. In each member of the ensemble we

introduce an indicator function φ(u ∗ ;

;

t) . It is so-named because it indicates when

u(t) < u ∗ (Figure 13.1) :

φ(u ∗ ; 1if u(t) < u ∗ ;

=

t)

(13.3)

φ(u ∗ ; t) = 0if u(t) ≥ u ∗ .

For any given t this set of indicator functions in effect counts th e ensemble
members in which u(t) < u . Put another way, its ensemble mean φ(u t) is the

fraction of the realizations in which u(t) < u . This is defined as the probability
distribution function P(u

;

; t) :

P(u ; t) = φ(u ; t). (13.4)

It should be evident that P(u ; t) has the properties

P(u 1 ; t) ≤ P(u 2 ; t), u 1 ≤ u 2 ,

; =

lim
u →∞

P(u t) 1 , (13.5)

lim P(u ; t) = 0 .

u →−∞

The derivative of the probability distribution with respect to amplitude is called
the probability density :

∂P(u ; t)

β(u ; t) ≡ . (13.6)

∂u