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Probability density function

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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.

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