Probability density function

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Stochastic methods use distribution functions to decribe the fluctuacting scalars in a turbulent field.

The distribution function $F_\phi(\Phi)$ of a scalar $\phi$ is the probability $p$ of finding a value of $\phi < \Phi$

$F_\phi(\Phi) = p(\phi < \Phi)$

The probability of finding $\phi$ in a range $\Phi_1,\Phi_2$ is

$p(\Phi_1 <\phi < \Phi_2) = F_\phi(\Phi_2)-F_\phi(\Phi_1)$

The probability density function (PDF) is

$P(\Phi)= \frac{d F_\phi(\Phi)} {d \Phi}$

where $P(\Phi) d\Phi$ is the probability of $\phi$ being in the range $(\Phi,\Phi+d\Phi)$ . It follows that

$\int P(\Phi) d \Phi = 1$

Integrating over all the possible values of $\phi$ , $\Phi$ is the sample space of the scalar variable $\phi$ . The PDF of any stochastic variable depends "a-priori" on space and time.

$P(\Phi;x,t)$

for clarity of notation, the space and time dependence is dropped. $P(\Phi) \equiv P(\Phi;x,t)$

From the PDF of a variable, one can define its $n$ th moment as

$\overline{\phi}^n = \int \phi^n P(\Phi) d \Phi$

the $n = 1$ case is called the "mean".

$\overline{\phi} = \int \phi P(\Phi) d \Phi$

Similarly the mean of a function can be obtained as

$\overline{f} = \int f(\phi) P(\Phi) d \Phi$

Where the second central moment is called the "variance"

$\overline{u'^2} = \int (\phi-\overline{\phi}) P(\Phi) d \Phi$

For two variables (or more) a joint-PDF of $\phi$ and $\psi$ is defined

$P(\Phi,\Psi;x,t) \equiv P (\Phi,\Psi)$

where $\Phi \mbox{ and } \Psi$ form the phase-space for $\phi \mbox{ and } \psi$ . The marginal PDF's are obtained by integration over the sample space of one variable.