Characterizations of the exponential distribution by order statistics and conditional expectations of record values

摘要

Let $ X_1 , X_2 , cdots , X_n$ be $n$ independent and identically distributed random variables with continuous cumulative distribution function $ F(x).$ Let us rearrange the $X's$ in the increasing order $ X_{1:n} le X_{2:n} le cdots le X_{n:n}$. We call $X_{k:n} $ the k-th order statistic. Then $X_{n:n} -X_{n-1:n}$ and $X_{n-1:n}$ are independent if and only if $F(x)=1- e^{-rac{x}{c}} $ with some $ c>0 $. And $ X_j $ is an upper record value of this sequence if $ X_j > max{ X_1 , X_2 , cdots , X_{j-1} }.$ We define $ u(n) =min {j ert j > u(n-1) , X_j > X_{u(n-1)} , n geq 2 } $ with $ u(1) =1.$ Then $ F(x)=1-e^{-rac{x}{c}}{hskip-0.05cm}, , x>0 $ if and only if $ E[X_{u(n+3)} - X_{u(n)} ert X_{u(m)} =y ] = 3c ,$ or $ E[X_{u(n+4)} - X_{u(n)} ert X_{u(m)} =y ] = 4c , , ,, n geq m+1 .$