# Probability density function

### From CFD-Wiki

m |
(Corrected the variance equation.) |
||

(7 intermediate revisions not shown) | |||

Line 26: | Line 26: | ||

\int P(\Phi) d \Phi = 1 | \int P(\Phi) d \Phi = 1 | ||

</math> | </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. | 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. | ||

+ | <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>. |

## 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 of a scalar is the probability of finding a value of

The probability of finding in a range is

The probability density function (PDF) is

where is the probability of being in the range . It follows that

Integrating over all the possible values of , is the sample space of the scalar variable . The PDF of any stochastic variable depends "a-priori" on space and time.

for clarity of notation, the space and time dependence is dropped.

From the PDF of a variable, one can define its th moment as

the case is called the "mean".

Similarly the mean of a function can be obtained as

Where the second central moment is called the "variance"

For two variables (or more) a joint-PDF of and is defined

where form the phase-space for . The marginal PDF's are obtained by integration over the sample space of one variable.

For two variables the correlation is given by

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

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

where is the conditional PDF.

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

and the mean value of a scalar can be expressed

only if and 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.

Finally a joint PDF of scalars
is defined as

where is the sample space of the array .