takehomestat4710 - 1 U in the usual manner i.e L 3 2 1 = =...

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University of Missouri Department of Statistics Introduction to Mathematical Statistics: 4710/7710 Take Home Examination Tuesday Oct 18, 2005 Return this exam by: 16:00 Thursday Oct 20, 2005 Examinations turned in after this time will not be graded. Student Name: Student Number: Sequence: Please mark your sequence with a cross: MWF 9:00-9:50 TR 9:30 – 10:45 TR 12:30 – 13:45 Instructions: 1. Attempt all questions. 2. Show all workings: Points will not be awarded unless workings are shown. 3. Students are reminder of the clause on academic honesty in the syllabus document. 4. Submitting this exam for assessment implies that you have conformed to the ethics and standards outlined in the clause on academic honesty, i.e. that the work submitted is your own honest effort.
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Question 1 [10] The continuous random variable X follows a uniform distribution ) 1 , 0 ( U . (a) Write down the density function for X . (2) (b) Show that the moment generating function for X is 0 , 1 ) ( = t t e t m t . (5) (c) Is this moment generating function useful for generating the moments of
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Unformatted text preview: ) 1 , ( U in the usual manner, i.e. L , 3 , 2 , 1 ), ( ) ( ) ( = = k m X E k k ? Explain. (3) Question 2 [20] A farmer performs quality inspection on each batch of 144 eggs packed for market. Suppose that in a particular batch there are 15 substandard eggs. Of course, this is not known to the farmer. He selects, without replacement, a dozen eggs at random, and rejects the batch if he finds three or more substandard eggs in his sample. Let X denote the number of substandard eggs in his sample. (a) Find the density for X . Hint: Make a sketch! = ) ( x f (3) Sketch: (3) (b) Find ) ( X E and ) Var( X ) ( X E = (2) ) Var( X = (2) (c) Set up the calculation needed to find ) 3 ( ≥ X P ) 3 ( ≥ X P = (2) (d) Use the binomial tables to approximate ) 3 ( ≥ X P ) 3 ( ≥ X P = (3) (e) What is the probability that the farmer will reject this batch of eggs? (1) (f) Criticize the farmers decision rule: (4) END...
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takehomestat4710 - 1 U in the usual manner i.e L 3 2 1 = =...

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