TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...

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Solution Note that in this problem, a random sample is meant an i.i.d. sample. a. ¯Xc is an unbiased estimate of the population mean µ iff µ = E( ¯X n∑ c )= c i E(X i ) =( i=1 n∑ c i )µ. i=1 This implies that the condition for unbiasness is n∑ c i =1. i=1 12
. Let Var(X 1 )=σ 2 . Then by the independence of X 1 ,...,X n , we have Var( ¯X n∑ c )=Var( c i X i ) = i=1 n∑ c 2 i Var(X i) i=1 n = σ 2 ∑ c 2 i . i=1 We want to minimize ∑ n ∑ i=1 c 2 i subject to ni=1 c i =1. To do that we minimize the following Lagrangian function: n∑ f(c 1 ,...,c n ,λ)= c 2 n i + λ(1 − ∑ c i ). i=1 i=1 Here λ is the Lagrangian multiplier. 13
Page 1 and 2: TUTORIAL 2 SOLUTIONS #7.7.11 Consid
Page 3 and 4: #7.7.12 Consider simple random samp
Page 5 and 6: Since E(Ȳ 2 ) = Var(Ȳ )=σ2 /n, E
Page 7 and 8: This shows that n −1 N 2 s 2 is a
Page 9 and 10: Solution a. Recall that the sample
Page 11: #7.7.24 For a random sample of size
Page 15: By taking 2nd partial derivatives o

TUTORIAL 2 SOLUTIONS #7.7.11 Consider a population of size four ...

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