A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Summary Let X 1 ,…, X n be mutually independent non‐negative integer‐valued random variables with probability mass functions f i ( x ) > 0 for z = 0,1,…. Let E denote the event that { X 1 ≥ X 2 ≥…≥ X n }. This note shows that, conditional on the event E, X i ‐ X i + 1 and X i + 1 are independent for all t = 1,…, k if and only if X i ( i = 1,…, k ) are geometric random variables, where 1 ≤ k ≤ n ‐1. The k geometric distributions can have different parameters θ i , i = 1,…, k.