## The Annals of Statistics

- Ann. Statist.
- Volume 9, Number 4 (1981), 822-833.

### On Distributions Determined by Random Variables Distributed Over the $n$-Cube

#### Abstract

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 the simplex is obtained. This result yields as a special case the distribution of the internally studentized extreme deviate; the cases of normal and exponential samples are considered in detail and the required distributions obtained.

#### Article information

**Source**

Ann. Statist., Volume 9, Number 4 (1981), 822-833.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176345522

**Digital Object Identifier**

doi:10.1214/aos/1176345522

**Mathematical Reviews number (MathSciNet)**

MR619285

**Zentralblatt MATH identifier**

0475.60012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 62E15: Exact distribution theory 62G30: Order statistics; empirical distribution functions

**Keywords**

Geometric probability order statistics linear combinations internally studentized deviate

#### Citation

Currie, Iain D. On Distributions Determined by Random Variables Distributed Over the $n$-Cube. Ann. Statist. 9 (1981), no. 4, 822--833. doi:10.1214/aos/1176345522. https://projecteuclid.org/euclid.aos/1176345522