Two numbers will be randomly (and independently) selected from a uniform distribution on an interval with length L and center M.

It is very easy to estimate M (just take the average of the two values), but I'm not sure how to construct a statistically reliable confidence interval for M based on the two values.

One the other hand, while it is not as obvious how to estimate L from the two values, I found a couple of different ways to construct statistically reliable confidence intervals for L, using the sample statistic X-Y. Indeed, here are two 99% confidence intervals for L.

• $\left[\frac{10}{9} |X-Y|, \infty\right)$
• $\left[|X-Y|, \frac{|X-Y|}{1 - \sqrt{.99}}\right]$

Two numbers will be randomly (and independently) selected from a uniform distribution on an interval with length $L$ and center $M.$

It is very easy to estimate $M$ (just take the average of the two values), but I'm not sure how to construct a statistically reliable confidence interval for $M$ based on the two values.

One the other hand, while it is not as obvious how to estimate $L$ from the two values, I found a couple of different ways to construct statistically reliable confidence intervals for $L,$ using the sample statistic $X-Y.$ Indeed, here are two $99\%$ confidence intervals for $L.$

• $\left[\frac{10}{9} |X-Y|, \infty\right)$
• $\left[|X-Y|, \frac{|X-Y|}{1 - \sqrt{.99}}\right]$