Probability density function

(Difference between revisions)

Revision as of 12:12, 17 October 2005

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 $\phi$ . The PDF of any stochastic variable depends "a-priori" on space and time.

$P(\Phi;x,t)$

@@ 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>