# Probability density function

### From CFD-Wiki

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

The distribution function of a scalar is the probability of finding a value of

The probability of finding in a range is

The probability density function (PDF) is

where is the probability of being in the range . It follows that

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

for clarity of notation, the space and time dependence is dropped.

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

the case is called the "mean".

Similarly the mean of a function can be obtained as

Where the second central moment is called the "variance"

For two variables (or more) a joint-PDF of and is defined

where form the phase-space for . The marginal PDF's are obtained by integration over the sample space of one variable.

For two variables the correlation is given by

This term often appears in turbulent flows the averaged Navier-Stokes (with ) and is unclosed.

Using Bayes' theorem a joint-pdf can be expressed as

where is the conditional PDF.

The conditional average of a scalar can be expressed as a function of the conditional PDF

and the mean value of a scalar can be expressed

only if and are correlated.

If two variables are uncorrelated then they are statistically independent and their joint PDF can be expressed as a product of their marginal PDFs.

Finnally a joint PDF of scalars
is defined as

where is the sample space of the array .