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I'm working a problem and I'm stuck on one part. Consider, X and Y, two independent discrete random variables who have the same geometric pmf. Show that for all n ≥ 2, the PMF

P(X=k|X+Y=n) is uniform.

Now, this equals: P(X=k)P(Y=n-k)/P(X+Y=n), which follows from the definition of conditional probability. Since the X and Y have the same geometric pmf the numerator is easy to calculate, but I'm stuck on what exactly P(X+Y=n) is. I know it's the joint PMF, but how can I relate it to the problem (i.e. to the fact that X and Y have the same geo PMF, that X and Y are independent etc). Any help is appreciated.

Thanks,

Alex