不規則騒音のレベル変動に関する統計的研究(III) : 不規則騒音の条件付確率密度分布陽表現

抄録

As shown in the previous paper, the bivariate joint probability density destribution P(E_1, E_2) of random noise can be generally expressed in the form of statistical Laguerre's orthonormal expansion series (cf. Eqs. (1) and (2)), where individual characteristics in the statistical properties of random noise (e. g. , the usual linear correlation and the general correlations of high order between two random noise variables) are reflected in four parameters m, s, m_2, s_2, and the expansion coefficients B(n_1, n_2) of joint probability density distribution. Letting m_1=m_2=m and using Eq. (3), the expansion expression of joint moment &ltE^(l_1)_1, E^(l_2)_2)&gt(l_1, l_2=1, 2) and the expansion coefficient B(n_1, n_2)(n_1, n_2=1, 2) are derived from Eq. (1) in a concrete manner by referring to Eqs. (7), (8), (9), (13), (14), and (15). However, for the purpose of putting the theory to practical use, we had better find an approximate expression of the bivariate joint probability density distribution in the closed form instead of using Eq. (1). Thus, we can derive an explicit expression of joint gamma probability distributions (31) and (32) in the closed form as a solution of the integral equation (18). Under the conditions (37), (38) and (39), a general expansion expression (1) in the form of statistical Laguerre's series can be approximated by Eqs. (31) and (32) in the closed form. The conditional probability density distribution of Bessel type derived from Eq. (32) can be rewritten in terms of Eqs. (47) and (48). Finally, the frequency distribution calculated from instantaneous readings of a soundlevel meter recorded at every five seconds (cf. JIS Z 8731) is shown in Fig. 2. And detailed experimental considerations of street noise enough to corroborate the above theoretical results are given in the following two cases:(a)the conditions (37), (38) and (39) for the possibility of approximating Eq. (1) to the closed form (32) (cf. Table 1). (b)the conditional probability density distribution of Bessel type in the form of Eqs. (47) and (48) (cf. Fig. 3).