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Unformatted text preview: 36-226 Summer 2010Homework 3Solutions1.X1,...,Xnare iid U(0,2). ThenfX(x) =12I[0,2](x)L() =1(2)nI[0,2](x1,...,xn)Note that you can not find the mle just by direct calculation in this case (which you canusually realize if you actually try solving it that way). So, you have to find the mle byinspection, like we did in class for the case of theU(0,) distribution.First, you note that in generalL() decreases asincreases ( (2)n 1(2)n).Also, we need to look at the values for whichis non-zero. SinceL() =1(2)nI[0,2](x1,...,xn) =1(2)nI[0,2](x(n))L() is non-zero ifxi2, i= 1,...,n2x(n)(the maximum)x(n)2.So, the likelihoodL() looks like this:X(n)2is the MLE.Now, we also need to find the expectation of the mle.1E[b] =EX(n)2=12E[X(n)] =To find the expectation of the maximum, we first find its pdf using the result about thedistribution of order statistics....
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This document was uploaded on 07/14/2011.
- Summer '09