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MASTER'S COMPREHENSIVE EXAMINATION Part I Theory (90 minutes) April 7th, 2012 Instructions: Attempt any three out of the following four questions on...

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MASTER’S COMPREHENSIVE EXAMINATION Part I Theory (90 minutes) April 7 th , 2012 Instructions: Attempt any three out of the following four questions on both pages. Mention which three you would like to have graded. Open book and open notes. 1. Let 1 2 , X X be independent with density / 2 ( ; ) , 0. x X e x f x x β β β - = > a) Find the MLE of β . b) Find the MLE of 1 . e β + 2. An urn initially contains 1 red ball and 1 blue ball. At each stage a ball is randomly withdrawn and replaced by two other balls of the same color. (For instance, if the red ball is initially chosen, then there would be 2 red and 1 blue ball in the urn when the next selection occurs.) Let , n i P be the probability that there are exactly i red balls in the urn after n stages have been completed 1 1. i n ≤ ≤ + a) Derive an equation for , n i P in terms of 1, , n j P - with boundary conditions. b) Calculate 9,1 9,2 and . P P 3. Let X be a single observation from a population whose density is 0 1 ( ;0) 0 , 0 1 0, otherwise X f x x x - = < < = Find the most powerful text of size α =.05 for 0 1 : 0 1 vs :0 2 H H = = 4. There are two bins that contain two different types of batteries. When in use, batteries in bin i last (in hours an exponentially distributed time with rate , 1,2. i i λ = Three batteries are independently chosen from the bins, each with probability i p from bin 1 2 , 1. i p p + = If all batteries are still operating after t hours of use, what is the probability that all three will still be operating after an additional s hours?
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a). The joint distribution is given by x1,x2 &gt;0 Taking log with base e we get L = =
Thus differentiating L with respect to β we get, Setting we get β=
The 2nd order differentiation at the...

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