On a Characterization of the Exponential Distribution by Conditional Expectations of Record Values

摘要

Let $ X_1 , X_2 , cdots $ be a sequence of independent and identically distributed random variables with continuous cumulative distribution function $ F(x).$ $ 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}}$, $x>0 $ if and only if $E[X_{u(n+1)} - X_{u(n)}$ $ert X_{u(m)}$ $=y ] =c $ or $E[X_{u(n+2)} - X_{u(n)}$ $ert X_{u(m)} =y ] = 2 c, n geq m+1.$