Probability density function
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.
Finally a joint PDF of scalars is defined as
where is the sample space of the array .