Two Interesting Properties of the Exponential Distribution

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