# ON A CHARACTERIZATION OF THE EXPONENTIAL DISTRIBUTION BY CONDITIONAL EXPECTATIONS OF RECORD VALUES

• Published : 2001.04.01

#### Abstract

Let X$_1$, X$_2$, … be a sequence of independent and identically distributed random variables with continuous cumulative distribution function F(x). X(sub)j is an upper record value of this sequence if X(sub)j > max {X$_1$, X$_2$, …, X(sub)j-1}. We define u(n) = min {j│j > u(n-1), X(sub)j > X(sub)u(n-1), n $\geq$ 2} with u(1) = 1. Then F(x) = 1 - e(sup)-x/c, x > 0 if and only if E[X(sub)n(n+1) - X(sub)u(n)│X(sub)u(m) = y] = c or E[X(sub)u(n+2) - X(sub)u(n)│X(sub)u(m) = y] = 2c, n $\geq$ m+1.

#### References

1. Characterization of the exponential distibution by some properties of the record vulues M. Ahsanuallah
2. Record Statistics
3. Elementary differential equations and boundary value problems E. Boyce;C. Diprima
4. Zeit. Wshrscheinlichkeitsth v.12 An outstanding value in a sequence of random variables M. N. Tata