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Probability density function

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(Corrected the variance equation.)
 
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\int P(\Phi) d \Phi = 1
\int P(\Phi) d \Phi = 1
</math>
</math>
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Integrating over all the possible values of <math> \phi </math>.
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Integrating over all the possible values of <math> \phi </math>,
 +
<math> \Phi </math> is the sample space of the scalar variable <math> \phi </math>.
The PDF of any stochastic variable depends "a-priori" on space and time.  
The PDF of any stochastic variable depends "a-priori" on space and time.  
:<math> P(\Phi;x,t) </math>
:<math> P(\Phi;x,t) </math>
 +
for clarity of notation, the space and time dependence is dropped.
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<math>  P(\Phi) \equiv P(\Phi;x,t) </math>
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 +
 +
From the PDF of a variable, one can define its <math> n </math>th moment as
 +
 +
:<math>
 +
\overline{\phi}^n = \int \phi^n  P(\Phi) d \Phi
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</math>
 +
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the  <math> n = 1 </math> case is called the "mean".
 +
 +
:<math>
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\overline{\phi} =  \int \phi P(\Phi) d \Phi
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</math>
 +
 +
Similarly the mean of a function can be obtained as
 +
 +
:<math>
 +
\overline{f} = \int f(\phi) P(\Phi) d \Phi
 +
</math>
 +
 +
Where the second central moment is called the "variance"
 +
 +
:<math>
 +
\overline{u'^2} = \int (\phi-\overline{\phi})^2 P(\Phi) d \Phi
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</math>
 +
 +
For two variables (or more) a joint-PDF  of <math> \phi </math> and <math> \psi </math> is defined
 +
:<math> P(\Phi,\Psi;x,t) \equiv P (\Phi,\Psi) </math>
 +
 +
where  <math> \Phi \mbox{ and }  \Psi </math> form the phase-space for
 +
<math> \phi \mbox{ and }  \psi </math>.
 +
The marginal PDF's are obtained by integration over the sample space of one variable.
 +
:<math>
 +
P(\Phi) = \int P(\Phi,\Psi) d\Psi
 +
</math>
 +
 +
For two variables the correlation is given by
 +
 +
:<math>
 +
\overline{\phi' \psi'} = \int (\phi-\overline{\phi}) (\psi-\overline{\psi}) P(\Phi,\Psi) d \Phi d\Psi
 +
</math>
 +
 +
This term often appears in turbulent flows the averaged Navier-Stokes (with <math> u, v </math>) and is unclosed.
 +
 +
Using Bayes' theorem a joint-pdf can be expressed as
 +
:<math>
 +
P(\Phi,\Psi) = P(\Phi|\Psi) P(\Psi)
 +
</math>
 +
where  <math> P(\Phi|\Psi) </math> is the conditional PDF.
 +
 +
The conditional average of a scalar  can be expressed as a function of the
 +
conditional PDF
 +
:<math>
 +
<\phi | \Psi > = \int  \phi P(\Phi|\Psi) d \Phi
 +
</math>
 +
and the mean value of a scalar can be expressed
 +
 +
:<math>
 +
\overline{\phi} = \int <\phi | \Psi > P(\Psi) d \Psi
 +
</math>
 +
only if <math> \phi </math> and <math> \psi </math> 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.
 +
:<math>
 +
P(\Phi,\Psi)= P(\Phi) P(\Psi)
 +
</math>
 +
 +
 +
Finally a joint PDF of <math> N </math> scalars <math> (\phi_1,\phi_2, ...,\phi_N) </math>
 +
is defined as
 +
:<math>
 +
P(\underline{\psi}; x,t) \equiv P(\underline{\psi})
 +
</math>
 +
where <math> \underline{\psi} =  (\psi_1,\psi_2, ...,\psi_N) </math> is the sample space of the array
 +
<math> \underline{\phi} </math>.

Latest revision as of 16:03, 20 May 2011

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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