涉及离散变量的函数统计矩点估计法

POINT ESTIMATE FOR THE FUNCTION WITH DISCRETE RANDOM VARIABLES

点估计法对于仅包含连续随机变量的函数和系统的随机分析具有原理简洁清晰、操作简单易行的优点,并可以直接给出除均值和标准差之外的其他低阶统计矩。然而,对于客观存在的或者是需处理为的涉及离散随机变量的系统,现有的点估计法无能为力。为解决这一问题,该文基于一般随机系统的形式解析解,导出了涉及离散变量函数和系统的统计矩估计的理论表达式;然后,将其与现有的点估计法相结合,给出了涉及离散变量的函数和系统的低阶矩估计的点估计法;最后,通过理论推导和算例分析两种方式验证了建议方法的合理性和有效性,且指出该方法对包含离散变量的一般工程随机系统分析的适用性。

The point estimate method is one of the simplest and most efficient methods for calculating statistical moments of stochastic systems or functions which only involve continuous random variables. However, when discrete random variables exist, inherently or artificially, existing point estimate methods are no longer suitable. To solve this problem, this paper presents a point estimate method for calculating statistical moments of stochastic systems or functions involving discrete random variables. The proposed method is deduced in two steps: the theoretical formulas for statistical moments of a stochastic system with discrete variables are derived based on the formal analytical solution of the general stochastic system, then the point estimate for a stochastic system or function with discrete variables are developed by introducing the existing point estimate for conditional statistical moment. At last, the rationality and validity of this method is verified by theoretical analysis and numerical examples involving two elementary mathematical functions and one single degree of freedom system. Results indicate that the proposed method provide an accurate estimates of statistical moments and show the potential of the proposed method for stochastic analysis of general engineering structures.