Final_Exam_Solution - Final Exam Solution ORIE 3500/5500:...

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Final Exam Solution ORIE 3500/5500: Engineering Probability and Statistics II Question 1 (i) (a) Probability Law: For a statistical experiment with sample space Ω and collection of events F , a probability law is a function P : F → R such that i. P ( A ) 0 for any event A , ii. P (Ω) = 1, and iii. P ( A 1 A 2 ... ) = P ( A 1 ) + P ( A 2 ) + ··· for disjoint events A 1 ,A 2 ,... . (b) Estimator: If X 1 ,...,X n are the available data to estimate an unknown parameter θ , then a function g ( X 1 ,...,X n ) is an estimator for θ . (c) Unbiased estimator: An estimator ˆ θ is unbiased for the parameter θ if E θ ( ˆ θ ) = θ θ. (d) Maximum likelihood estimator: If X 1 ,...,X n are the data with joint PDF (or PMF) L ( θ ) = f X ( x 1 ,...,x n ), then the MLE is an estimator ˆ θ which maximizes L ( θ ) over θ . (ii) (a) M Z ( s ) = E [ e s ( a ( X - 1)+ bY ) ] = E [ e saX e - sa e sbY ] = e - sa E [ e saX ] E [ e sbY ] = e - sa M X ( sa ) M Y ( sb ) . (b) ˆ
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Final_Exam_Solution - Final Exam Solution ORIE 3500/5500:...

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