The Measurement of Conditional Probability Distribution Function for Random Energy Fluctuations

We need, very often, to measure the joint probability distribution of many correlative random physical quantities in case when we observe simultaneously one or more waves having random phases (like random noise) at two or more different observation points that are different in time, position, frequency or propagation constant. If we call our particular attention to the effect of general correlations among their random physical quantities, it would be well to measure directly the conditional probability distribution rather than the joint probability distribution.This paper proposes a new method for obtaining sample points of the conditional probability distribution by using the combination of Schmitt trigger circuit and phantastron. We can vary the trigger level in Schmitt trigger circuit and the delay time by use of variable condensers at will. And then, this paper gives a series of photographs prepared to illustrate directly the effect of delay time τ on the wave form of output fluctuation by intensity modulation. Three explicit expressions of conditional probability density distribution for white noise, after passing a mean squaring circuit and an audio band-pass filter of arbitrary width, are experimentally derived in connection with an autocorrelation function of input noise. These are reasonably expressed in terms of Bessel-, Biased Γ-, and Γ-types of distribution.In this case, the conditional probability distribution for the above output fluctuation is theoretically treated as a probability problem of “distance” in an N-dimensional function space with N=2TW (W: frequency interval, T: time interval), where dsitance is taken as the mean squared fluctuation.The experimental and theoretical results described in this paper are also applicable to the other fields of measurement on random phenomena, since the mean energy is a universal physical quantity.