This preview shows page 1. Sign up to view the full content.
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 equallylikely 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 longrun 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� longrun+ 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 discretetime 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...
View
Full
Document
This document was uploaded on 03/19/2014 for the course EECS 6.436 at MIT.
 Fall '08
 DavidGarnarnik

Click to edit the document details