Sufficient Statistics Homework Statement and Solution

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Homework Statement


Problem is in the attachment, sorry, I can't figure out how to do tex in this message board system.


Homework Equations





The Attempt at a Solution


In the attachment.
 

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I think you're into too much math. The I(.) function, if I get its meaning correctly, is nothing but to indicate that it is impossible to sample x\leq\mu. With frequentists' approach, this means a lot: you can safely ignore the bunch of I(.)'s and consider only those samples greater than \mu. This should solve the problem :wink:
 
abeliando said:

Homework Statement


Problem is in the attachment, sorry, I can't figure out how to do tex in this message board system.


Homework Equations





The Attempt at a Solution


In the attachment.

Ignoring the fancy notation, what you have for given, fixed μ is that Y = X-μ is exponentially distributed with unknown mean, and you are trying to estimate that mean. You are being asked to show some property of the "usual" estimate.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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