TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...
TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...
TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...
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a. Let µ = E(X 1 ) and Y i = X i −µ. Then<br />
E(Y i )=0,<br />
Var(Y i )=Var(X i )=σ 2 ,<br />
E(s 2 )= 1 n∑<br />
E[X<br />
n − 1 i − µ − ( ¯X − µ)] 2<br />
i=1<br />
= 1 n∑<br />
E(Y<br />
n − 1 i − Ȳ )2<br />
i=1<br />
= 1 n∑<br />
E(Y 2<br />
n − 1<br />
i − 2Y iȲ + Ȳ 2 )<br />
i=1<br />
= nσ2<br />
n − 1 −<br />
+ n<br />
n − 1 E(Ȳ 2 )<br />
= nσ2<br />
n − 1 −<br />
2n<br />
n − 1 E(Ȳ 2 )<br />
n<br />
n − 1 E(Ȳ 2 ).<br />
4