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# mid2a - 332:321 Probability and Stochastic Processes...

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332:321 Probability and Stochastic Processes Examination 2 November 22, 2004 You have 80 minutes to answer the following questions in the notebooks provided. You are permitted 1 double-sided sheet of notes. Make sure that you have included your name, Rutgers netid and signature in each book used. (5 points) Read each question carefully. All statements must be justified. Computations should be simplified as much as possible. The following table may be helpful: x 0 . 0 0 . 25 0 . 5 0 . 75 1 . 0 1 . 25 1 . 50 1 . 75 2 . 0 2 . 25 2 . 50 ( x ) 0 . 50 0 . 599 0 . 692 0 . 773 0 . 841 0 . 894 0 . 933 0 . 960 0 . 977 0 . 988 0 . 994 1. 40 points SHORT ANSWER: Each part is a separate problem. (a) Y is a Gaussian = 2 , σ = 2 ) random variable. Calculate P [ Y > 3]. (b) X 1 , X 2 and X 3 are independent identically distributed (iid) continuous uniform random variables. Random variable Y = X 1 + X 2 + X 3 has expected value E [ Y ] = 0 and variance σ 2 Y = 9. What is the PDF f X 1 ( x ) of X 1 ? (c) X is a Gaussian = 0 , σ = 1 ) random variable. Z is a Gaussian ( 0 , 4 ) random variable. X and Z are independent. Let Y = X + Z . Find the correlation coefficient ρ of Z and Y . Are Z and Y independent?
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