Two Interesting Properties of the Exponential Distribution

TL;DR

This paper investigates properties of the correlation coefficients between order statistics of exponential distributions, revealing their monotonic behavior and comparing them with uniform distributions, along with a combinatorial identity.

Contribution

It establishes the increasing and decreasing patterns of correlation coefficients among exponential order statistics and compares these with uniform distributions, providing new insights into their dependence structure.

Findings

Correlation coefficient between $X_{(k)}$ and $X_{(k+t)}$ is strictly increasing then decreasing in $k$.

Correlation between $X_{(1)}$ and $X_{(n-1)}$ exceeds that between $X_{(2)}$ and $X_{(n)}$.

Correlation coefficients for exponential variables are always less than those for uniform variables.

Abstract

Let $X_1, X_2,\ldots, X_n$ be $n$ independent and identically distributed random variables, here $n \geq 2.$ Let $X_{(1)}, X_{(2)}, \ldots, X_{(n)}$ be the order statistics of $X_1, X_2,..., X_n.$ In this note we proved that: (I) If $X_1, X_2,..., X_n$ are exponential random variables with parameter $c > 0,$ then the "correlation coefficient" between $X_{(k)}$ and $X_{(k+t)}$ is strictly increasing in $k$ from $1$ to $m,$ and then is strictly decreasing in $k$ from $m$ to $n - t,$ here $t$ is a fixed integer between $1$ and $n - 3,$ and $m = (n - t)/2$ if $n - t$ is even, $m = (n - t + 1)/2$ if $n - t$ is odd. We also proved that if $t = n - 2$, then the "correlation coefficient" between $X_{(1)}$ and $X_{(n-1)}$ is greater than the "correlation coefficient" between $X_{(2)}$ and$X_{(n)}.$ (II) The "correlation coefficient" between $X_{(k)}$ and $X_{(k+t)}$ for the exponential random variables is always less than the "correlation coefficient" between $X_{(k)}$ and $X_{(k+t)}$ for the uniform random variables for all $k$ and $t$ such that $k + t \leq n.$ A combinatorial identity is also given as a bi-product. \vs