Stationary probability density functions: An exact result

An exact expression is obtained for the probability density function (pdf) of any quantity measured in a general stationary process, in terms of conditional expectations of time derivatives of the signal. This expression indicates that the conditional expectations of both the time derivative squared and of the second time derivative influence the shape of the pdf, including its tails. A previous result of Ching [Phys. Rev. Lett. 70, 283 (1993)] for temperature measurements in turbulent flows corresponds to the particular case when the latter quantity is linear.