Probability density function

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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 $\underline{\phi}$ .

Probability density function

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@@ Line 6: / Line 6: @@
 :<math>
-F_\phi(\Phi) = p(phi < \Phi)
+F_\phi(\Phi) = p(\phi < \Phi)
 </math>
@@ Line 13: / Line 13: @@
 :<math>
-p(\Phi_1 <phi < \Phi_2) = F_\phi(\Phi_2)-F_\phi(\Phi_1)
+p(\Phi_1 <\phi < \Phi_2) = F_\phi(\Phi_2)-F_\phi(\Phi_1)
 </math>
@@ Line 26: / Line 26: @@
 \int P(\Phi) d \Phi = 1
 </math>
-Integrating over all the possible values of <math> \phi </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>.