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Poisson processes

I have a proof question regarding poisson processes. I know that you can prove that in a poisson process, the interarrival times follow an exponential distribution, and you can prove that the arrival times are uniform. I am wondering how you would prove this to be true for the discrete case, if the interarrival times are geometric, prove that the arrival times are discrete uniform, i.e. a discrete poisson process.

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