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Suppose $X_1$ and $X_2$ are taken at random from a uniform distribution on the interval $[\theta - 1/2, \theta + 1/2]$, where $\theta$ is unknown $(-\infty < \theta < \infty)$. Let $Z = Y_2 - Y_1$, where $Y_1 = min(X_1, X_2)$ and $Y_2 = max(X_1, X_2)$.

How do I calculate the conditional distribution of $X = 0.5(X_1 + X_2)$ given $Z=z$? Specifically, how do I show that this conditional distribution is uniform on the interval $[\theta - 1/2(1 - z), \theta + 1/2(1 - z)]$?

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    You can use $\TeX$ on this site by enclosing formulas in dollar signs; single dollar signs for inline formulas and double dollar signs for displayed equations. You can see the source code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". [Here](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference)'s a basic tutorial and quick reference. There's an "edit" link under the question.2012-09-27

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