Week-6-Midterm-Review.pdf - LECTURE MIDTERM REVIEW GROUP 1...

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LECTURE: MIDTERM REVIEW GROUP 1
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Study Diary Week 6 Random Processes 1 Problems 1.1 Problem 1 We have: f(x) = 0.1 - 0.005x We want to calculate: P ( X > 16 | X > 8) = P ( X > 16 , X > 8) P ( X > 8) = P ( X > 16) P ( X > 8) = R 20 16 f ( x ) dx R 20 8 f ( x ) dx = 1 - 0 . 96 1 - 0 . 64 = 1 9 1.2 Problem 5 Let D, R, I denote the states of a voter, respectively Democrat, Republican and independent. (a) P = D R I D 2 / 3 0 1 / 3 R 0 4 / 5 1 / 5 I 1 / 2 1 / 2 0 (b) As P k > 0 (for some k 1), P is a regular matrix. Thus, there exists one unique limiting distribution π = ( π 1 π 2 π 3 ) Now, we calculate π : πP = π 2 3 π 1 + 1 2 π 3 = π 1 4 5 π 2 + 1 2 π 3 = π 2 1 3 π 1 + 1 5 π 2 = π 3 π 1 + π 2 + π 3 = 1 π 1 = 0 . 3 π 2 = 0 . 5 π 3 = 0 . 2 (c) P 3 = D R I D 0 . 5185 0 . 2444 0 . 237 R 0 . 1466 0 . 672 0 . 1813 I 0 . 3555 0 . 4533 0 . 1911 P ( X 3 = R | X 0 = R ) = 0 . 672 1.3 Problem 9 (a) P ( N 90 = 0) = P ( N 1 90 = 0 , N 2 90 = 0) = P ( N 1 90 = 0) P ( N 2 90 = 0) = e - 0 . 02 * 90 * (90 * 0 . 02) 0 0! * e - 0 . 03 * 90
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