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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 . The PDF of any stochastic variable depends "a-priori" on space and time.


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


and the marginal PDF's are obatined by integration over the sample space of one variable.

P(\Phi) = \int P(\Phi,\Psi) d\Psi

For two variables the correlation is given by

\overline{\phi' \psi'} = \int (\phi-\overline{\phi}) (\psi-\overline{\psi}) P(\Phi,\Psi) d \Phi d\Psi

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

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

P(\Phi,\Psi) = P(\Phi|\Psi) P(\Psi)

where  P(\Phi|\Psi) 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.

P(\Phi,\Psi)= P(\Phi) P(\Psi)
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