On the restrictiveness of the hazard rate order

09/04/2020

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by   S. Fried, et al.

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Every element θ=(θ_1,…,θ_n) of the probability n-simplex induces a probability distribution P_θ of a random variable X that can assume only a finite number of real values x_1 < ⋯ < x_n by defining P_θ(X=x_i) = θ_i, 1≤ i ≤ n. We show that if Θ and Θ' are two random vectors uniformly distributed on Δ^n, then P(P_Θ≤_ hr P_Θ')=1/2^n-1 where ≤_ hr denotes the hazard rate order.