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exam_1_m4320_f2010_v1_0

# exam_1_m4320_f2010_v1_0 - Exam 1 Math 4320 Fall 2010...

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Exam 1: Math 4320 Fall 2010 Problem 1. (12) Let ( X n ) n =0 be a Markov chain with state space { 0 , 1 , 2 } , initial probability distribution P ( X 0 = i ) = a i for 0 6 i 6 2, and transition matrix M = P 00 P 01 P 02 P 10 P 11 P 12 P 20 P 21 P 22 . Express the following probabilities in terms of the a i and the P ij . (a) (4) P ( X 1 = 2 , X 2 = 1 | X 0 = 1) (b) (4) P ( X 1 = 2 , X 2 = 1) (c) (4) P ( X 0 = 0 , X 1 = 0 , X 2 = 1) Problem 2. (16) Let X be a continuous random variable with probability density function (pdf) f X . (a) (4) Deﬁne the expected value E [ X ]. (b) (4) Deﬁne the variance Var[ X ]. (c) (8) Suppose that f X ( t ) is given by f X ( t ) = ± 3 /t 4 , if t > 1; 0 , if t < 1 . Compute E [ X ] (4 points) and Var[ X ] (4 points). Problem 3. (10) Let ( ξ i ) i =1 be a sequence of independent, identically distributed random variables and let N be a discrete random variable with range { n Z : n > 0 } . Deﬁne the random sum S by S = ± 0 , if N = 0; n i =1
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