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Let Yi,Yz Yabe iid Uniform(0, 0) random variables_ where > 0 is an Unknown parameter. Note that this itplics that E(Y;) 0/2. aId V(Y;) = 0*/12 for cach (10 point...

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Let Yi,Yz Yabe iid Uniform(0, 0) random variables_ where > 0 is an Unknown parameter. Note that this itplics that E(Y;) 0/2. aId V(Y;) = 0*/12 for cach (10 points) Let "E" Prove that T, is consistent estimator of 0.(10 points) Suppozt tuatt Lt max{Yi,Y. Ywo} icte largest of Y1 Yi: cAn be showu that the cuulative distribution fuuction (cdf) o M isI <0 F(r) = P(M < =) = (2/0)1 , 0 <f < 0 r >0Prove that 0,M/(0.10!)) is 907 confideuce interval for 0.

Let Yi,Yz Yabe iid Uniform(0, 0) random variables_ where > 0 is an Unknown parameter. Note that this itplics that E(Y;) 0/2. aId V(Y;) = 0*/12 for cach (10 points) Let "E" Prove that T, is consistent estimator of 0. (10 points) Suppozt tuatt Lt max{Yi,Y. Ywo} icte largest of Y1 Yi: cAn be showu that the cuulative distribution fuuction (cdf) o M is I <0 F(r) = P(M < =) = (2/0)1 , 0 <f < 0 r >0 Prove that 0,M/(0.10!)) is 907 confideuce interval for 0.



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Suppose $X_{1}, X_{2}, \ldots, X_{n}$ are iid with pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty$, zero elsewhere. Find the mle of $P(X \leq 2)$ and show that it is consistent.


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