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conditional distribution

Source: LIMIT 2019 CCS2 P4

April 28, 2021
probability

Problem Statement

Let X,YX,Y be i.i.d Geom⁡(p)\operatorname{Geom}(p). What is the conditional distribution of X∣X+Y=kX|X+Y=k? <spanclass=′latex−bold′>(A)</span> Uniform⁡{1,2,…,⌊k2⌋}<span class='latex-bold'>(A)</span>~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor\right\} <spanclass=′latex−bold′>(B)</span> Uniform⁡{1,2,…,k}<span class='latex-bold'>(B)</span>~\operatorname{Uniform}\left\{1,2,\ldots,k\right\} <spanclass=′latex−bold′>(C)</span> Uniform⁡{1,2,…,⌊k2⌋+1}<span class='latex-bold'>(C)</span>~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor+1\right\} <spanclass=′latex−bold′>(D)</span> None of the above<span class='latex-bold'>(D)</span>~\text{None of the above}
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