Probability density function

2011-05-20T16:03:25Z

Willings: Corrected the variance equation.

Stochastic methods use distribution functions to decribe the fluctuacting scalars
in a turbulent field.

The distribution function <math> F_\phi(\Phi) </math> of a scalar <math> \phi </math> is the probability
<math> p </math> of finding a value of <math> \phi < \Phi </math>

:<math>
F_\phi(\Phi) = p(\phi < \Phi)
</math>

The probability of finding <math> \phi </math> in a range <math> \Phi_1,\Phi_2 </math>
is

:<math>
p(\Phi_1 <\phi < \Phi_2) = F_\phi(\Phi_2)-F_\phi(\Phi_1)
</math>

The probability density function (PDF) is

:<math>
P(\Phi)= \frac{d F_\phi(\Phi)} {d \Phi}
</math>

where <math> P(\Phi) d\Phi </math> is the probability of <math> \phi </math> being in the range <math> (\Phi,\Phi+d\Phi) </math>. It follows that
:<math>
\int P(\Phi) d \Phi = 1
</math>
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.
:<math> P(\Phi;x,t) </math>
for clarity of notation, the space and time dependence is dropped.
<math> P(\Phi) \equiv P(\Phi;x,t) </math>

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
</math>

the <math> n = 1 </math> case is called the "mean".

:<math>
\overline{\phi} = \int \phi P(\Phi) d \Phi
</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
</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>.

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Probability density function