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Suppose we have a uniform random variable $X$ (continuous) in an interval of length one $[\gamma,\gamma+1]$ but we don't know $\gamma$. We just take one sample $x_1$.

What can we say about $\gamma$? Can we give an estimate of its value?

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That $x_1-1\leqslant\gamma\leqslant x_1$. Estimated value: $\gamma=(x_1-\frac12)\pm\frac12$.

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That's it? Can we assign probabilities to the points in $\gamma=(x_1-\frac12)\pm\frac12$? –  Ambesh Aug 8 '13 at 9:50
    
"That's it?" Obviously, nothing more can be deduced from one observation. "Can we assign probabilities to the points..." The question is unclear, you might want to explain. –  Did Aug 8 '13 at 10:58
    
What I mean if it's more likely that $\gamma=x_1$ than $\gamma=x_1-1$. I somehow see that it looks more likely $\gamma=x_1$ and it becomes less likely as we move away. But it's just my intuition. –  Ambesh Aug 8 '13 at 13:28
    
Wrong. If you must give one value, choose $\gamma=x_1-\frac12$. This is the value which minimizes the error (and any $\gamma=x_1-\delta$ with $0\lt\delta\lt1$ would be better than $\gamma=x_1$). –  Did Aug 8 '13 at 13:36
    
Could you justify these statements? –  Ambesh Aug 8 '13 at 18:57
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