抄録
The Bessel type distribution worked out in our previous paper is not a very practical expression as an explicit expression of the conditional probability in the field of random street noises, since it is expressed mathematically in a complicated form. For the purpose of introducing a truly practical expression of the conditional probability density, it is advisable to find an approximate expression in a more simplified form instead of using the above Bessel type distribution. Thus, we can find out that a lognormal probability distribution can be approximately calculated as a universal expression of the conditional probability distribution of random noises. First, a joint moment generating function of noise variables Z_h (h=l, 2) defined by a logarithm of random noise level X_h (db) is given from a general expression of the bivariate joint probability in the form of statistical Laguerre expansion series. Then, the 2-dimensional normal distribution can be approximately derived as an expression of the bivariate joint probability density P(Z_1, Z_2) in the closed form. Accordingly, we can easily obtain an explicit expression of the conditional probability density P(X_1 X_2) given by a lognormal distribution, where information of linear correlation between two random noise levels X_1 and X_2 is reflected in the parameter of the conditional average. If our attention is focussed on the shape of a conditional probability distribution irrespective of the internal structure of the linear correlation effect reflected in parameters, a lognormal probability distribution can be more simply found from another point of view. Finally, detailed experimental considerations of street noises sufficing to corroborate the above theoretical results are given in the following two cases:(i) the conditional probability density function P(X_1 X_2) in the form of a lognormal distribution, (ii) the detection of weak periodical signals buried in random street noises.
