On Distributions Determined by Random Variables Distributed Over the $n$-Cube
由 n 维立方体上随机变量确定的分布
The distribution function of a random variable of the form $\sum^n_{i = 1} a_i Y_1 Y_2 \cdots Y_i$ where $a_i > 0$ and $0 \leq Y_i \leq 1$ is considered. A geometric argument is used to obtain the distribution function as a repeated integral. The result is used first to obtain the distribution function of a linear combination of variables defined over the simplex $X_i \geq 0, \sum^n_{i = 1} X_i \leq 1$. As a second application the distribution of certain quadratic forms over…