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Unformatted text preview: MS&E 221 Midterm Examination
Ramesh Johari February 13, 2006 Instructions 1. Take alternate seating if possible. 2. Answer all questions in the spaces provided on these sheets. If needed, additional paper will
be available at the front of the room. Answers given on any other paper will not be counted. 3. The examination begins at 1:05 pm, and ends at 2: 15 pm.
4. No notes, books, or calculators are allowed.
5. The exam will be scored out of 150 points. 6. Pace yourself! Note that part 2(f) and part 3(e)(f) are harder than the rest of the exam, and
you should leave them for last. Honor Code In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) 50 L U T10 N 3 NAME (printed) Problem 1 (40 points). Answer each of the following questions TRUE or FALSE. If TRUE, provide a brief justiﬁcation
(1—2 sentences) for your answer; if FALSE, provide a brief counterexample. (5 points per correct
answer; 5 points per correct justiﬁcation/counterexample) (a) Every Markov chain with a ﬁnite state space has at least one closed class. Titus. I4, mt, all clasu are n01‘ closed =5 all
slates are akansc'euf =9 i EIJiofvm‘k ioiixo‘i] (a =’ ZE[#avi.%4ule.rk]<p V k, 3“ 3'04: 44‘: chain M9 Mal“: NIH} Man: with
40 «I sillu, width is intranet“. w (1 (M1! shift 6F“; (b) Every Markov chain on a ﬁnite state space with a symmetric transition matrix has an invariant
distribution. (A matrix is symmetric if Aij = Ajl for all i, j.) —9 ““5“ I; P is 53mm, 11' is doubl; .ylwluxh‘c)
So 4*! warm Amman :5 «want (m 9.5.2) (c) If 71' is an invariant distribution for a Markov chain, and 7m > 0 and 7rj > 0, then 11 H j. FALSE. 93“ {5' (I o) (d) If a Markov chain is irreducible and the mean return time to a state i is ﬁnite, then a unique invariant distribution exists. (The mean return time to state 7'. is E[R,(1)XU = 7'], where
Ri(1) =111in{n 2 1 : Xn =i}.) TWE I? thztlﬂxo'il <w, am e is raw/m
recurrent; sine 44“ do?” is I'nreduciblc,
all skie5 are rsthva vewmnt
VasiJ'ive WWW =9 invariant JWW‘W exck'h'. ﬁrducwmiy =9 MWi' 443%me h unto)“: . Problem 2 (50 points). (Double or nothing gambling) A gambler has $71 (where 1 g i, g 5), and wants to turn it into $6.
At each stage, the gambler chooses a wager, and a fair coin is tossed (i.e., the coin has probability
1/2 of coming up heads, and probability 1/2 of coming up tails). If the coin comes up heads, the
gambler wins an additional amount equal to his wager. If the coin comes up tails, the gambler loses
his wager. He follows a strange strategy: if his current wealth is an odd number of dollars, he wagers $1. If his current wealth is an even number of dollars, he wagers $2. He stops playing either when his
wealth reaches $0 (bankrupt), or $6. (a) (10 points) Give a Markov chain description of the wealth of the gambler, with state space {0,...,6}. _.
l ‘ c
i l /"_\/——\g
.M 9"ng 0‘7?
0W vi 3 ‘ t 5 r i ‘ I
l
'i I 1mm: WM: Ptx.=z) = 1. (b) (10 points) What are the communicating classes of the Markov chain? Which classes are
closed? Ww‘bakna dug: ' i0?
fl}
22,4“:
13} i5}
is} Only f0) and {63 av! cbmd‘ (C) (10 points) Give the following limit for each i, () g i S 6: lim P(X,, = ilXU : 1') 11—490 Since all (=; 0,6 are haiwf,
3‘32 chn'elﬁ’il = 0 ‘4' °<‘<(°' Since ( ¢ 0, 6 axe absorbing;
£5: wxuaz lX,=Cl =1 .4 no .. M». (d) (10 points) Give the following limit for each i, () g i S (5: lllll P(X,) : ()‘iAXU : I) MM Making («[04an 
p,‘ ‘ PUm‘l' e if. =0 —l
“5’,E*5lhq
,I n ,2
,9“ i‘whaahﬁ'i
:7 k2=§
3i =9
=9 hI=éth3 3.! hi 2
\'e, “:13. (e) (10 points) (Harder; complete qfter the rest of the exam!) Now suppose the gambler will
keep playing (using the same strategy) until his wealth reaches either $0 or $21x', for some
positive integer K. Give the following limit: liin P(X,, : 2Kan : 3) Il—'OO k; e [’(M 2KX°=C) =ll$.\—~
1k LL
.3.
2K Problem 3 (60 points). (Data transmission) A computer has a network interface that can be in two states: idle or
transmitting. At each time period that the interface is idle, with probability p (0 < p < 1) a data
packet arrives; in this case, the interface enters the transmitting state. In the transmitting state, no data packets airive. Instead, at each time period in the transmitting
state, the interface attempts to transmit the given data packet through the network: each attempted
transmission is successful with probability (1 (0 < q < 1), and otherwise fails. If a transmission is successful, the interface returns to the idle state. If a transmission fails, the
interface remains in the transmitting state, up to a maximum of two transmissions for a given data
packet; after two failed transmissions, the interface returns to the idle state (we say the data packet
was lost). Assume the interface is currently idle. (a) (10 points) Give a Markov chain description for the state of the interface. 191.6 f. 1 (1 1 1: about h "'3 Far o.—————’ OM. OW hmm‘m‘m.
te ‘l 1 2‘abw‘l' ¥WKW MMSM“5.M ' PD“ ’1) 3" (b) (10 points) Is this Markov chain irreducible? Are all states aperiodic? I! {; INWtblc‘ I is aft/wool?“ 159in M I "1_”2 a: “we 3.c.o{.1 5° d1 shad(J 0N Art/violin“:  (C) (10 points) Suppose a packet has just arrived to the transmitter. What is the probability the
packet is lost? “MW“: at gum t.»  ((7)1 (d) (10 points) What is the probability that 41+ least three packets are successfully transmitted
before the ﬁrst packet loss? Pmblbalf‘ly a racial Ls .suausl’ullg {muMHed= 1— (#1?
Once a rum is succex ML” lrammrlki, 4w
mm valorb sz pug. Wm“
Gym :5va 4w rwbubfli'l’ « fatal; ava/es 8 1.. «1 least’
50 " fwlvabtlt'b 01‘ «444% smear»! wsm‘n’im ‘ ( l (HF): (e) (10 points) (Harder; complete after the rest aft/1e exam!) What is the expected number of
packets that will be lost before the ﬁrst successful packet transmission? . This 1MAR¢M is tubing Jw 44v. MW «number of
Ms 1» t: befvre +w {353+ mlum lo 5 in in Lumaj Marker chain' S\\:‘\/\i_ M [5“ «Med 7 7 t—f JN’JJ'] (f) (10 points) (Harder; complete after the rest thhe exam!) Now suppose that if a transmission
fails, the interface waits T time periods to transmit. where [’(T : n) : r” "(l — r),
for 71 Z 1. (If T = l, the interface attempts transmission at the next time period.) You
should assume that wait times after each failure are independent and identically distributed.
Describe the state of the interface as a Markov chain. I“? 7'
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