General formula for stationary or statistically homogeneous probability density functions

A general formula is derived for the probability density function (PDF) of fluctuating physical quantities measured in any stationary or statistically homogeneous process. For stationary processes, the formula relates the PDF to two conditional means: two averages involving a general function of the quantity and its time derivatives, the time derivative of this function and the time derivative of the quantity, taken when the fluctuating quantity is at a certain value. A previous result by Pope and Ching [Phys. Fluids A 5, 1529 (1993)] is a special case of this general formula when the function is chosen to be the time derivative of the fluctuating quantity. An analogous formula is obtained for the PDF of fluctuating physical quantities measured in statistically homogeneous processes with spatial derivatives in place of time derivatives.