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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 ,  \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})^2 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.


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.

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


<\phi | \Psi > = \int  \phi P(\Phi|\Psi) d \Phi

and the mean value of a scalar can be expressed


\overline{\phi} = \int <\phi | \Psi > P(\Psi) d \Psi

only if  \phi and  \psi 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.


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


Finally a joint PDF of  N scalars  (\phi_1,\phi_2, ...,\phi_N) is defined as


P(\underline{\psi}; x,t) \equiv P(\underline{\psi})

where  \underline{\psi} =  (\psi_1,\psi_2, ...,\psi_N) is the sample space of the array  \underline{\phi} .

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