On a Characterization of the Exponential Distribution by Conditional Expectations of Record Values
Commun. Korean Math. Soc. 2001 Vol. 16, No. 2, 287-290
Min-Young Lee
Dankook University
Abstract : 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 $u(n)$ $=$ $\min$ $\{j \vert j$ $>$ $u(n-1)$, $X_j > X_{u(n-1)}$, $n \geq 2 \}$ with $ u(1) =1.$ Then $ F(x)=1-e^{-\frac{x}{c}}$, $x>0 $ if and only if $E[X_{u(n+1)} - X_{u(n)}$ $\vert X_{u(m)}$ $=y ] =c $ or $E[X_{u(n+2)} - X_{u(n)}$ $\vert X_{u(m)} =y ] = 2 c, n \geq m+1.$
Keywords : absolutely continuous distribution, characterization, record value
MSC numbers : 60E15, 62E10
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