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Unformatted text preview: Math 540/640: Statistical Theory I Final Exam: Fall 2009 Instructor: Songfeng (Andy) Zheng Note: This final exam is designed for you to review the introduced topics and apply the learned skills to expand your knowledge domain. If necessary, some background information is given, and all the problems can be solved using the skills you learned from class. You have 24 hours to finish the problems, please try to solve the problems without referring to the notes and textbook as possible as you can. Please show your work in detail as much as you can. Please write your solutions clearly, use larger font size so that I can read your work! Problem 1: Basic calculations. Let Y 1 and Y 2 have a joint density function as f ( y 1 ,y 2 ) = 3 y 1 , ≤ y 2 ≤ y 1 ≤ 1 and f ( y 1 ,y 2 ) = 0 for other values of y 1 and y 2 . a. Please find the marginal density functions of Y 1 and Y 2 . b. Find P ( Y 1 ≤ 3 / 4 | Y 2 ≤ 1 / 2). c. Find the conditional density function of Y 1 given Y 2 = y 2 . d. Find P ( Y 1 ≤ 3 / 4 | Y 2 = 1 / 2). e. Find Cov(Y 1 , Y 2 ). f. Find the correlation coefficient between Y 1 and Y 2 . Problem 2: Moment generating function and Poisson distribution. Let random variables X 1 , ··· ,X n be independent Poisson distributed, and X i has mean value λ i > 0. a. Please use the definition to calculate the moment generating function of X i . [ Please show your calculating process, if you just give the answer, you will get 0! ] b. Find the moment generating function of X 1 + X 2 + ··· + X n , and thus determine its probability function. c. Please find the conditional distribution of X 2 + X 5 , given that X 1 + X 2 + ··· + X n = m ....
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This note was uploaded on 11/17/2010 for the course MTH 540 taught by Professor George during the Summer '09 term at Missouri State University-Springfield.
- Summer '09