# Let X 1 , X 2 , · · · , X n represent a random sample...

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1 Statistics 323 Sample Final Exam 1. Let X 1 , X 2 , · · · , X n represent a random sample of values taken on a population variable having the following probability density function f ( x | α , β ) = 1 Γ ( α ) β α x α 1 e x/ β for 0 < x 0 elsewhere. (a) For known value of α > 0, find a maximum likelihood estimator for β . [5] Copyright Jim Stallard 2016
Statistics 323 Sample Final Exam: Copyright Jim Stallard 2016 2 (b) Find a method of moments estimator for β . [5] Copyright Jim Stallard 2016
Statistics 323 Sample Final Exam: Copyright Jim Stallard 2016 3 2. Let X 1 , · · · , X n and Y 1 , · · · , Y n represent two random samples of size n , each taken from a distinct population where X is normally distributed with a mean μ X and variance σ 2 X ; Y is normally distributed with a mean μ Y and a variance σ 2 Y . If σ 2 X = σ 2 Y = σ 2 , is the estimator: S 2 p = n i =1 ( X i X ) 2 + n j =1 ( Y j Y ) 2 2 n 2 a consistent estimator for σ 2 ? Completely justify your answer. [10] Copyright Jim Stallard 2016
Statistics 323 Sample Final Exam: Copyright Jim Stallard 2016 4 3. Let X 1 , X 2 , · · · , X n be a random sample taken from a population which has a random behavior described by the following probability density function: f ( x ) = 1 θ 0 x θ (a) Is X max (or X ( n ) ) an unbiased estimate 1 for θ ? If not suggest an unbiased estimator for θ . [2] (b) A random variable W has the following density: f ( w ) = nw n 1 0 w 1