This is actually part of a problem from All of Statistics:

$X_1, X_2, \ldots, X_n \sim \text{Uniform}(0, \Theta)$. And $Y = \text{Max}\{X_1,\ldots, X_n\}$.

If you're given that $Y > c$, can you estimate the probability of $\Theta>1/2$?

Of course if $c\ge1/2$ the probability is 100%.

Any hint or direction is appreciated.

BTW, is there anywhere I can find answers to the book All of Statistics? It's really a good book except there's no solution to the problems, even part of them.

  • $\begingroup$ If $\Theta$ is random, and $Y>c$ for one sample, then $P(\Theta>1/2)$ is not necessarily 1 if $c>1/2$. Could you state the problem in its entirety? Now it is not clear what does it mean that $Y>c$. Does it mean that $P(Y>c)=1$ or that for given sample $Y>c$. $\endgroup$ – mpiktas Oct 21 '11 at 3:15

First find the distribution of $Y$ for a given $\Theta$. Then you need to define a prior for $\Theta$. Then calculate the joint distribution of $Y$ and $\Theta$. Then integrate to get Pr($Y > c$ and $\Theta>1/2$) and Pr($Y>c$), and divide. Note that you are calculating the probability rather than estimating it.


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