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# HW10 - θ If the average of a sample of 10 batteries is 36...

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OR 3500/5500, Summer’09, Chen Homework 10 Due on Wednesday, June 24, 3pm. Problem 1 (a)Find the maximum likelihood estimator of the unknown parameter θ where X 1 ,X 2 ,...,X n is a sample from the distribution whose density function is f X ( x ) = e - ( x - θ ) if x θ 0 otherwise. (b)Find the moment estimator of the parameter θ in (a). Problem 2 Based on the MLE estimator in part (a) of Problem 1, construct a one-sided upper conﬁdence interval for the unknown parameter at conﬁdence level 1 - α . For α = . 05, what is the smallest sample size that will result in the conﬁdence interval at most . 01 long? Problem 3 Suppose the lifetimes of batteries are exponentially distributed with mean
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Unformatted text preview: θ . If the average of a sample of 10 batteries is 36 hours, determine the 95 percent two-sided and both one-sided conﬁdence intervals for θ . The following problem is optional for 3500 students, mandatory for students enrolled in 5500 Problem 4 Suppose X 1 ,...,X n are i.i.d. from normal distribution N ( μ,σ 2 ). Let ¯ X n = ∑ n i =1 X i n (sample mean) and S 2 = ∑ n i =1 ( X i-¯ X n ) 2 n-1 (sample variance). Show that ¯ X n and S 2 are independent random variables. Hint: use the pdf transformation formula of the random vectors and the Jacobian is just some constant. 1...
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