# intro theory of statistics 1

please finish the questions without any programming.

3)LetX, Y be two continuous random variables with joint density function
8 xy / q 4if0 < y < x< q
f(x,yÂ½q)=

otherwise .

We want to estimate q relative to Mean Square Error (MSE).From lecture we already know that X is a sufficient statistic forq and has density function
fX (xÂ½q)=4 x3 / q4 if0 < x < q
0 otherwise
Consider the estimatord (x,y) =3 x y. Without computing any MSE find an estimator that will have smaller MSE than the estimatord (x,y)=3 x y and explain why that should be true.
4)We have one observation,Y,having pdf, for -1â‰¤qâ‰¤1, equal to
2 y q + 1if-1/2<y<1/2
f (yÂ½q)=
0otherwise.
NOTE:For f (yÂ½q) to be a density function the only possible values ofq must be in the closed interval -1â‰¤ qâ‰¤1.
Consider the estimator W(y) defined as
W( y)= 1ifyÂ³0
-1ify <0
a.IsW(y)an unbiased estimator ofq?
b..Show that W(y) is an MLEofq.

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