Unformatted text preview: X 1 ,X 2 ,...,X n following N ( μ,σ 2 ) with μ known but σ 2 unknown. In this case, ∑ n i =1 ( X iμ ) 2 /σ 2 is a valid pivot and can be used to ﬁnd a conﬁdence interval for σ 2 . (2 × 5 = 10 points). (2) Consider the model Y i = β X i + ² i for i = 1 , 2 ,...,n . Here the X i ’s are ﬁxed constants and the ² ’s are i.i.d N (0 ,σ 2 ) random variables. (a) Are the Y i ’s independent and identically distributed in this case? Explain. (b) Find the M.L.E’s of ( β,σ ) ( 7 points). (3) Let X 1 and X 2 be two i.i.d. observations from an Exponential( λ ) distribution. Suppose X 1 = 1 . 8 and X 2 = 2 . 6. Find a conﬁdence interval for λ based on the above data. (6 points) Note: Tables of quantiles of standard distributions are available in the Appendix Section of Rice’s book. 1...
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 Winter '08
 moulib
 Statistics, Maximum likelihood, Likelihood function, Yi, Moulinath Banerjee, conﬁdence interval i=1

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