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Let X1 and X2 be the numbers on two independent fair-die rolls. Let X be the minimum and Y the maximum of X1 and X2. Calculate: $$E(Y|X=x)\qquad\text{and}\qquad E(X|Y=y) $$ given that X1 and X2 independent and uniformly distributed on $\{1,\ldots,n\}$.

What I was able to do was trying to rewrite $E(Y|X=x) = E(Y,X=x)/(E(X=x))$, but this does not seem to helpful. Also, what can I do with the fact that they are indpendent and uniformly distributed?

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    I would start by calculating the result for small $n$ and look for a pattern. For $n=2, E(Y|x=2)=2, E(Y|x=1)=\frac{5}{3}$ as the possibilities are (1,1), (1,2), and (2,1). E(X=i)=E(y=n-i) by symmetry.2011-04-27

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