CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Wiki > Probability density function

Probability density function

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
(Corrected the variance equation.)
 
(5 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
From the PDF of a variable, one can define its <math> n </math>th moment as
Line 51: Line 55:
:<math>
:<math>
-
\overline{u'^2} = \int (\phi-\overline{\phi}) P(\Phi) d \Phi
+
\overline{u'^2} = \int (\phi-\overline{\phi})^2 P(\Phi) d \Phi
</math>
</math>
For two variables (or more) a joint-PDF  of <math> \phi </math> and <math> \psi </math> is defined
For two variables (or more) a joint-PDF  of <math> \phi </math> and <math> \psi </math> is defined
-
:<math> P(\Phi,\Psi;x,t) </math>
+
:<math> P(\Phi,\Psi;x,t) \equiv P (\Phi,\Psi) </math>
-
and the marginal PDF's are obatined by integration over the sample space of one variable.
+
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>
:<math>
P(\Phi) = \int P(\Phi,\Psi) d\Psi
P(\Phi) = \int P(\Phi,\Psi) d\Psi
Line 75: Line 81:
</math>
</math>
where  <math> P(\Phi|\Psi) </math> is the conditional PDF.
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.
If two variables are uncorrelated then they are statistically independent and their joint PDF can be expressed as a product of their marginal PDFs.
Line 80: Line 98:
P(\Phi,\Psi)= P(\Phi) P(\Psi)
P(\Phi,\Psi)= P(\Phi) P(\Psi)
</math>
</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} .

Retrieved from "https://openfoam.cfd-online.com/Wiki/Probability_density_function"
My wiki