100BHW3 - STAT 100B Homework III Problem 1. Please derive...

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Unformatted text preview: STAT 100B Homework III Problem 1. Please derive the exponential distribution and interpret the meaning of the param- eter . Calculate E ( X ), E ( X 2 ), and Pr( X > t ). Problem 2. Continue. Let X 1 , X 2 , ..., X n be independent observations from Exp( ). Derive the estimators of from the following estimating equations: (1) E ( X ) = ( X 1 + ... + X n ) /n . (2) E ( X 2 ) = ( X 2 1 + ... + X 2 n ) /n . (3) Pr( X > t ) = n i =1 1 X i >t /n for fixed t (lets use t = 1). Suppose X 1 , X 2 , ..., X n are 0.5399 1.4874 1.5651 0.5659 0.8786 0.0296 1.5420 0.8868 0.3808 1.5744, where n = 10. Please calculate the estimates using the above three estimators. Problem 3. Continue. I generate 5 data sets from Exp(1), each data set consists of 10 observa- tions: (1) 0.5399 1.4874 1.5651 0.5659 0.8786 0.0296 1.5420 0.8868 0.3808 1.5744 (2) 0.8592 0.5451 0.9681 0.2304 1.1874 0.0100 0.4408 0.2950 1.5485 0.4989 (3) 0.6626 0.2740 0.2442 2.8271 0.1343 0.2372 1.1393 1.3170 0.1753 0.4622(3) 0....
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