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hw5 - θ θ 1 4 Consider a random sample of size 72 from...

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APPM 4/5520 Problem Set Five (Due Wednesday, October 12th) 1. Suppose that X 1 , X 2 , . . . , X n iid N ( μ, σ 2 ). (a) Show that n i =1 ( x i - μ ) 2 σ 2 = n i =1 ( x i - x ) 2 σ 2 + n ( x - μ ) 2 σ 2 . (b) Write down the joint pdf for X 1 , X 2 , . . . , X n and use part (a) to rewrite the “ e -exponent part”. (c) Consider the joint transformation Y 1 = X , Y i = X i - X for i = 1 , 2 , . . . , n . Use the Jacobian method to find the joint pdf for Y 1 , Y 2 , . . . , Y n . Show that Y 1 is inde- pendent of Y 2 , . . . , Y n . (d) Show that X 1 - X = - n i =2 ( X i - X ). Conclude that X 1 - X is independent of X . (e) Conclude that X is independent of the sample variance S 2 . 2. Suppose that X 1 , X 2 , . . . , X n iid N ( μ, σ 2 ). (a) Find the distribution of n i =1 ( X i - μ ) 2 σ 2 . (b) Find the distribution of n ( X - μ ) 2 σ 2 . (c) Use problem 1a and moment generating functions to find the distribution of ( n - 1) S 2 σ 2 . 3. Let Y 1 , Y 2 , . . . , Y n denote a random sample from the Beta ( θ, 1) distribution. Show that Y is a consistent estimator of
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Unformatted text preview: θ/ ( θ + 1). 4. Consider a random sample of size 72 from the distribution with pdf f ( x ) = 1 x 2 I (1 , ∞ ) ( x ). Compute the approximate probability that more than 50 of the observations are less than 3. 5. Suppose that a random sample of size 10, taken from the N ( μ, 2) distribution, results in a sample mean of 7 . 4. Give an 80% conFdence interval for the true mean μ . 6. Required for 5520 Students Only: Let X 1 , X 2 , . . . , X n be a random sample from any distribution with variance σ 2 . Assume that the Frst four moments, μ k = E [ X k ], k = 1 , 2 , 3 , 4, of this distribution exist. Show that S 2 P → σ 2 . where S 2 is the sample variance S 2 = ∑ n i =1 ( X i-X ) 2 n-1 ....
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