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Let $Y_{1}<Y_{2}<â€¦<Y_{n}$ be the order statistics of $n$ independent observations from a uniform distribution with p.d.f. $f(y)={Î¸1â€‹,0,â€‹0â‰¤yâ‰¤Î¸,Î¸>0Â elsewhereÂâ€‹$ (a) What is the p.d.f. of $Y_{n}$ ? (b) Find $E(Y_{n})$ and $E(Y_{n})$. Hence show that $Var(Y_{n})=(n+1)_{2}(n+2)nÎ¸_{2}â€‹$ (c) If $Î¸^=nn+1â€‹Y_{m}$ is the estimator of $Î¸$, show that $Î¸^$ is unbiased. (d) Compute $P(cÎ¸<Y_{n}<Î¸)$ for $0<c<1$. Use this result to construct a $95%$ confidence interval for $Î¸$ if $n=10$ and the observed value of $Y_{10}$ is $y_{10}=0.741$.

(a)

To find the probability density function (p.d.f.) of Y_n, we need to determine the cumulative di...

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