Abstract.    Given two independent positive random variables, under some minor conditions, it is known that from E(X^r | X + Y) = a(X+Y)^r and E(X^s | X+Y) = b(X+Y)^s, for certain pairs of r and s, where a and b are two constants, we can characterize X and Y to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t > 0} be a renewal process, with {S_k, k > 1} the sequence of arrival times, and F the common distribution function of the inter-arrival times. We prove that for some fixed n and k, k < n, if E(S^r_k | A(t) = n) = at^r and E(S^s_k | A(t) = n) = bt^s, for certain pairs of r and s, where a and b are independent of t, then {A(t), t > 0} has to be a Poisson process. We also give some corresponding results about characterizing F to be geometric when F is discrete.

Key words and phrases:    Characterization, exponential distribution, gamma distribution, geometric distribution, Poisson process, renewal process.

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