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For a random sample $X_1,X_2,...X_n$ from a uniform $[0,\Theta]$ distribution, with probability density function

$$f(x;\Theta) = \left\{ \begin{array} \ \frac{1}{\Theta} & 0\le x \le\Theta,\\ 0 & \text{otherwise}.\end{array}\right.$$

What is the value of $k$ such that $\hat{\Theta}=k\bar{X}$ is an unbiased estimator of $\Theta$?

I've done some questions similar to this but I'm not sure how to go about this one. I have a test in 3 hours so help is really appreciated!

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2 Answers 2

up vote 7 down vote accepted

$\overline{X}= \frac1n\sum X_i$ $\mathrm{E}(X_i)=\theta/2 =\mathrm{E}(\overline{X})$. So $\theta =\mathrm{E}(2\overline{X})$. Hence $k=2$. Joriki gave the correct answer but for the OP to understand an explanation seems in order. I hope this is in time for your test.

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2  
Hey thanks so much! A question I'd done before came up instead so I got lucky! Test went great thanks to everyone who answered my questions! :) –  Fred Aug 11 '12 at 12:46

$2{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

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16  
I would upvote, but I don't want to ruin the $2$ in the votes. :-) –  Asaf Karagila Nov 24 '12 at 20:50
16  
I'd upvote your comment, @Asaf, but.... –  Cameron Buie Dec 17 '12 at 15:50
16  
I'd upvote your comment @CameronBuie, but... –  user26698 Dec 30 '12 at 16:17
16  
I'd upvote your comment @user26698, but... –  Parth Kohli Jan 10 '13 at 8:24
16  
I also upvoted your comment @Novice, but... –  stalking is prohibited Jan 26 '13 at 14:32

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