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 . The PDF of any stochastic variable depends "a-priori" on space and time.
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
and the marginal PDF's are obatined 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.
If two variables are uncorrelated then they are statistically independent and their joint PDF can be expressed as a product of their marginal PDFs.