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find conditional distribution

Source: LIMIT 2019 CCS2 P11

April 28, 2021
probability

Problem Statement

Let X1,X2,X3X_1,X_2,X_3 be exp⁡(1)\exp(1). Find the conditional distribution of X1∣X1+X2+X3=kX_1|X_1+X_2+X_3=k. <spanclass=′latex−bold′>(A)</span> Uniform⁡(0,k)<span class='latex-bold'>(A)</span>~\operatorname{Uniform}(0,k) <spanclass=′latex−bold′>(B)</span> Uniform⁡(0,k3)<span class='latex-bold'>(B)</span>~\operatorname{Uniform}\left(0,\frac k3\right) <spanclass=′latex−bold′>(C)</span> Uniform⁡(0,2k3)<span class='latex-bold'>(C)</span>~\operatorname{Uniform}\left(0,\frac{2k}3\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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