Statistical Characteristics Of The Exponential Distribution And Geometric Distribution

We study some new characterizations of exponential distribution and geometric distribution in this paper. Here are the following results:(1) Characterizations of exponential distribution: Let X be a non-negative continuous random variable with a random sample of size2, X(1) and X(2)-X(1) are independent if and only if X has an exponential distribution. Furthermore, we discuss some other situations:Let X be a non-negative continuous random variable with a random sample of size n, X(1) and X(2)-X(1) are independent if and only if X has an exponential distribution; X(1) and X(k)-X(1) are independent if and only if X has an exponential distribution; X(1) and X(3)-X(2) are independent if and only if X has an exponential distribution; X(1) and X(k)-X(5) are independent if and only if X has an exponential distribution; X(1) and X(k)-X(s) are independent if and only if X has an exponential distribution.(2) Characterizations of geometric distribution:If X has an exponential distribution, then X(i) and X(k)-X(i) are not independent (k＞i＞1); TP2dependent of geometric distribution and RCSI of geometric distribution.