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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. – Ross Millikan Apr 27 '11 at 5:11

Your formula $E(Y|X=x)=E(Y,X=x)/(E(X=x))$ is not correct; in fact, $E(X=x)$ doesn't even make sense (events don't have expectations, they have probabilities).

I'd start by computing the conditional probability $P(X=x | Y=y)$ (note that the answer will be different depending on whether $x < y$, $x=y$ or $x > y$). Then, use the fact that conditional expectation can be computed just like ordinary expectation, replacing probability by conditional probability: $$E[X | Y=y] = \sum_{x=1}^n x P(X=x | Y=y).$$

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I got this down, but can you work out E(Y|X=x)? I'm lost on this. – mary Apr 28 '11 at 6:25

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