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Two independent observations $X_1$ and $X_2$ are made from a continuous random variable $X$ having probability density function $f(x)=\frac{1}{k}$, $0\leq x\leq k$.

Find the probability distribution of M, the larger of $X_1$ and $X_2$, and hence show that $\frac{3M}{2}$ is an unbiased estimator of $k$.

How do you find the probability distribution of M?

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Hint: $P(M \le m) = P(X_1 \le m, X_2 \le m)$. You should be able to take it from here. Use the fact that $X_i$ are independent random draws from the distribution of $X$ to express the above probability in terms of the CDF of $X$.

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  • $\begingroup$ I'm aware of your expression. I have obtained the probability for that, which is $\frac{m^2}{k^2}$, is the next step just to differentiate it? $\endgroup$ – user373534 Oct 8 '17 at 8:46
  • $\begingroup$ Yes as that is the CDF. Once you have the pdf you should be able to proceed from there to show that $M$ is an unbiased estimator for $k$. $\endgroup$ – Srikant Oct 8 '17 at 9:04
  • $\begingroup$ One more question, could you help me calculate $Var(\frac{3M}{2})$? I'm getting a negative number and that isn't correct. I know it's $\frac{9}{4}Var(M)$ but I'm still not getting a correct answer. EDIT: Scratch that, I've got it. Thanks. $\endgroup$ – user373534 Oct 8 '17 at 9:39
  • $\begingroup$ Post a separate question with the work you have done so far. $\endgroup$ – Srikant Oct 8 '17 at 9:41

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