The Distribution of Products of Independent Random Variables
The Distribution of Products of Independent Random Variables
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DOI：
10.1137/0114046
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发表时间：
1966-03
影响因子：
1.9
通讯作者：
M. Springer;W. E. Thompson
中科院分区：
文献类型：
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作者：
M. Springer;W. E. Thompson
MD弹簧和W。E.汤普森1.摘要基本方法的推导n个独立随机变量（irv的）的产品的概率密度函数（pdf的），并用于获得特定的结果，除了情况n 2，被认为是新的。该方法使用Mellin积分变换，并且是Epstein [1]提出的方法的n个变量的推广。对于n个单式1，n < 10柯西，n <7高斯变量的乘积，得到了Pdf的非显式形式.用这种方法计算了n 2，3，6高斯irv的N（0，1）的乘积表[12]，本文中包括了它的简化版本。未经删节的表格中的数据精确到小数点后六位，并允许在面积栏中进行线性插值，精确到四位数。这些表为从事可靠性分析、通信理论和其它需要考虑irv产品的应用的工程师和研究人员提供了一个迄今为止还没有的工具。导论.与irv的和的分布不同，两个以上的乘积的分布受到的关注相对较少，现有的结果对应用科学家提供的有用信息也很少。爱泼斯坦[1]提出了一个系统的方法来研究产品ofi。rv的使用了梅林积分变换，但没有对两个以上的积进行应用。Levy [2]提出了关于irv乘法的一般理论问题，并导出了两个变量乘积的一些结果。1962年，Zolotarev [3]在无穷可分分布的基础上，开始建立一个类似于加法理论的一般乘法理论。他的程序是在一系列定理中进行的，没有证明，这些定理既显示了与加法结果的相似性，也显示了与加法结果的不同。Jam-bunathan [4]和Sakamoto [5]分别导出了beta和矩形irv的乘积的分布。其他既定成果交易
MD SPRINGER AND W. E. THOMPSON 1. Summary. Fundamental methods are developed for the derivation of probability density functions (pdf’s) of products of n independent random variables (irv’s), and are used to obtain particular results which, aside from the case n 2, are believed to be new. The methods use the Mellin integral transform, and are a generalization to n variables of a method presented by Epstein [1]. Pdf’s are obtained inexplicit form for products of n monomial1, n-< 10 Cauchy, and n _< _ 7 Gaussian variables. Tables for products of n 2, 3, 6 Gaussian irv’s N (0, 1) have been calculated using this method [12], abridged versions of which are included in this paper. Entries in the unabridged tables were obtained with accuracy to six decimal places and permit linear interpolation with four-digit accuracy in the area column. These tables offer a heretofore unavailable tool to the engineer and research scientist concerned with reliability analysis, com-munications theory, and other applications requiringconsideration of products of irv’s.2. Introduction. Unlike the distribution of sums of irv’s, the distribution of products of more than two has received relativelylittle attention, and results which are available supply little useful information to the applied scientist. Epstein [1] has suggested a systematic approach to the study of products ofi. rv’s using the Mellin integral transform, but did not carry out the application for products of more than two. Levy [2] posed the question of a general theory of multiplication of irv’s and derived some results for products of two variables. In 1962, Zolotarev [3] began the con-struction ofa general theory of multiplication of irv’sanalogous to the theory of addition based on infinitely divisible distributions. His program was carried out in a sequence of theorems, stated without proof, which show both the similarity to, and difference from, the results for addition. Jam-bunathan [4] and Sakamoto [5], respectively, have derived the distribution of products of beta and rectangular irv’s. Other established results deal