hw1sol - 36-226 Summer 2010Homework 1Due June 301. Math

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Unformatted text preview: 36-226 Summer 2010Homework 1Due June 301. Math review.(a)lnnYi=1eabxixi!!=nXi=1lneabxixi!=nXi=1a+xilnb-xiXj=1lnj=na+ lnbnXi=1xi-nXi=1xiXj=1lnj(b)f(x|a,n) =alnx+ (n-a)ln(1-x)f(x|a,n)x=ax-n-a1-xax=n-a1-xa-ax=nx-axx=an(critical point)2f(x|a,n)x2=-ax-n-a(1-x)2<Therefore,anmaximizesf(x|a,n).(c)nXi=1(Xi-)2=nXi=1(Xi-X+X-)2=nXi=1((Xi-X) + (X-))2=nXi=1(Xi-X)2+ (X-)2-2(Xi-X)(X-)=nXi=1(Xi-X)2+nXi=1(X-)2-2(X-)nXi=1(Xi-X)=nXi=1(Xi-X)2+n(X-)-2(X-)=nXi=1(Xi-X)2+n(X-)12. Probability review.(a)E[X1] =1Zxexp{-x/}=1-xexp{-x/}+Zexp(-x/)(integration by parts)= 0 +=.Now recall thatV ar[X1] =E[X21]-E[X1]2, so we need to findE[X21].E[X21] =1Zx2exp{-x/}dx=1-x2exp{-x/}+ 2Zxexp{-x/}dx(integration by parts)= 0 + 22(using the same procedure as forE[X1])Therefore,V ar[X1] =E[X21]-E[X1]2= 22-2=2....
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hw1sol - 36-226 Summer 2010Homework 1Due June 301. Math

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