midterm - . Note that the distribution of X has pdf f X ( x...

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M4056 Midterm Exam, Oct. 13, 2010 Name Section I. Provide a precise mathematical defnition For each oF the Following basic notions oF mathematical statistics, and provide an example or illustration. (±ive points each.) 1. Bernoulli random variable 2. Moment generating Function 3. Chi squared random variable 4. Statistic 1
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5. Sufcient statistic 6. Method oF moments estimator 7. Maximum likelihood estimator 8. Mean squared error (oF an estimator) 2
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Section II. 1. [20 points] Let S 2 = S 2 ( X 1 , . . ., X n ) be the sample variance of a sample of size n from a population described by a random variable X with distribution f X ( x ). (I.e., the X i are independent, identically distributed, each with the same distribution as X .) Explain the diFerence in meaning between: a) the variance of X , b) the sample variance (i.e., S 2 itself), c) the expected value of S 2 , d) the variance of the sample mean (Var M ), e) the variance of S 2 (Var S 2 ). 3
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2. [15 points] Let ( X 1 , . . ., X n ) be a sample of size n from an exponential distribution with parameter λ
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Unformatted text preview: . Note that the distribution of X has pdf f X ( x | ) = e-x , 0 x . a) Write the sample distribution f v X ( vx | ). b) Find a sucient statistic for . c) What is the MLE of ? d) What is the mean squared error of the MLE? Section III. 1. [25 points] Suppose the random variables Y 1 , . . ., Y n satisfy: Y i = + x i + i , i = 1 , . . ., n, where x 1 , . . ., x n are xed constants, and 1 , . . ., n are iid normal(0 , 2 ), 2 unknown. a) Find a three-dimensional sucient statistic for ( , , 2 ). 4 b) Express the MLE of as a linear function of the statistics from part a) and . c) Express the MLE of as a linear function of the statistics from part a) and . d) Suppose 2 and are Fxed. What is the MLE of ? Is this estimator biased or unbiased? e) Suppose 2 and are Fxed. What is the MLE of ? Is this estimator biased or unbiased? 5...
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This note was uploaded on 11/29/2011 for the course MATH 4056 taught by Professor Staff during the Fall '08 term at LSU.

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midterm - . Note that the distribution of X has pdf f X ( x...

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