Question: Suppose X U ( 0 , ) be the uniform distribution on ( 0 , ) . Note that log p ( x , )

Suppose XU(0,) be the uniform distribution on (0,). Note that log p(x,) is differentiable for all >x, that is, with probability 1 for each . And we can thus define moments of T = / log p(X,) . Show that

E[/ log P(X,)] = 1=0
Var[/ log P(X,)] = 0 and I()=ET2 = Var T + (ET)2=1/2
2X is unbiased for and has variance (1/3)2(1/I())=2.

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