final_2007

# 60416431 probabilistic systems sum of independent c

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Unformatted text preview: ation Engineering & Computer Science is t/2. 6.041/6.431: Probabilistic Systems sum of independent (c) Male and female arrivals are independent processes, and the Analysis Poisson random variables is again Poisson. The answer is Poisson with parameter λ1 b + (Spring 2007) λ2 (t − a). (d) Since each arrival N1 + N2 independently comes from N1 with probability Oscar goes for a run each morning. When he leaves his house for his run, he is equally-likely to λ1 b p= go out either the front or the back door; λ1nd λ2 (t − a) , , when he returns, he is equally likely to a b + similarly go to either the front or back 1 conditional on N1owns=onlybinomial airsparameters n = 10 es which he takes the distribution of Ndo or. Oscar + N2 10 is ﬁve p with of running sho a fter the run t which oﬀ immediately and p. Its variance isa10p(1 − p).ever door he happens to be. If there are no shoes at the �t−1 (e) Supposeesisto go running, , t)) = uns barefo o1)]. The random variables door from which he leav t integer. Then, N ([0 he r i=0 N ([i, i + ted. We are interested in determining N ([i, i + 1)) are iid with ﬁnite variance of λ1 . Then, applying the central limit theorem the long-run proportion of time t)) ≈ Nhλ(tr− 1), λ(t − 1))oted. probability that it is above approximation, N ([0, that ( e uns barefo and the (a) (b) 867. its mean is approximately 1/2, by the symmetry of the normal distribution. If s not an a Markcan chain, sp ecifying the states and t where Set the scenariot iup as integer we ov make a similar argument by deﬁning Δt = t/�t�,ransition �t� is the largest integer smaller than t. Then Δt is between 1 and 2, and N ([0, t)) = �� Determine thet� long-run+ 1)Δt)), and theof time Oscarbefore applies. oted. proportion same argument as runs barefo i=0 N ([iΔt, (i probabilities. Problem 2: (23 points) 6_3_markov_steady_2_11.tex Markov chain shown in the ﬁgure. Consider the discrete-time 1/3 1 1/2 1/2 2 3 1 3/4 4 1/4 1/2 1/6 3/4 5 1/2 6 1/2 1/4 (a) (3 pts.) What are the recurrent classes? (b) (5 pts...
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## This document was uploaded on 03/19/2014 for the course EECS 6.436 at MIT.

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