exam_2_m4320_f2010_v1_0

# exam_2_m4320_f2010_v1_0 - (e) (3) P ( X (1) = 0 | X (1) =...

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Exam 2: Math 4320 Fall 2010 Problem 1. (10) Let ( X ( t )) t > 0 be a Poisson process with rate λ . Let S 1 be the time at which the ﬁrst event occurs. Suppose that at time T > 0 we have X ( T ) = 1. Show that conditioned on X ( T ) = 1, the conditional PDF of S 1 has the uniform distribution f S 1 | X ( T )=1 ( t ) = ± 1 /T, if 0 6 t 6 T ; 0 , otherwise . Problem 2. (10) Let ( X n ) n =0 be a branching process such that each organism produces ξ oﬀspring ( ξ is a nonnegative integer-valued random variable). Prove that for all integers n > 1, we have E [ X n ] = ( E [ ξ ]) n . Problem 3. (15) Let ( X ( t )) t > 0 be a Poisson process with rate λ = 2. Compute the following. (a) (3) P ( X (4) = 1) (b) (3) E [ X (4)] (c) (3) P ( X (1) = 1 and X (2) = 3) (d) (3) P ( X (2) = 3 | X (1) = 1)
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Unformatted text preview: (e) (3) P ( X (1) = 0 | X (1) = 2) Problem 4. (10) A certain manufacturing process produces tabletops with defects that occur according to a spatial Poisson process with a mean rate of 1 defect per tabletop. If 2 inspectors check separate halves of a given tabletop, what is the probability that both inspectors ﬁnd defects? Problem 5. (10) Let ( X ( t )) t > be a nonhomogeneous Poisson process with rate function λ ( s ) = ± s 2 , if 0 6 s < 1; 1 /s 2 , if s > 1 . (a) (5) Compute E [ X (1)]. (b) (5) Compute lim t →∞ E [ X ( t )] . 1...
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## This note was uploaded on 10/03/2011 for the course MATH 3364 taught by Professor Staff during the Spring '08 term at University of Houston.

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