Question

1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x € S = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Prz(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed (c) Suppose 1x = 1 is constant for all x E S. Prove by induction that Px-kx(t) = (at) ke Af/k! for k = 0,..., and x = 0,1,2,.... (d). Suppose 1 = 3 arrivals per minute. Find the probability that, given there were 5 arrivals in 7 minutes, the first arrival was in the first minute. (e) Suppose = 3 and that there have been 100 arrivals so far. What is the probability that in the next 4 minutes, there will be at most 1 arrival.

Answer 1

12 Chapman Volcanegaror Equation (fearande best Py(+1T). Po & given the state wasi at epoch of the state gat epoch tut, but passing from a I in time (HT), la puers through some state Kin timet. stute ito Pro (+1)== Prod XIt+T)=), X14) = * 181030] ER X24)=k 17.00)=c) Pry XC+1)=) | Aloyza,x 1)= Por *(741)=J1XC)=x H)=k] - Poş x 1t+1) =3 / XH)=k] alg Po (HT)= R02 (+). Px (I) P1+++) = Plt).PLT) ag = a Pg H/to; it) di Pau (+) 10 ay = 2t Pig (ot) - P2 (0) - Zir Peg (8t) ito at Po(t) = 0,47 +0(04); C)

an = zur Pr (8t) Piolo stod - 24 Piilott stost ! Piv (st) = lreau (at) + 0.184) agro, o it ari Lo ?? (4) 1 zagro 3 arg aan que quatre as are called Pandian semulates Ascan) R (HT) = 2 (HT) = Pop (4) a Rey (7) Po 'H) = 7 Pield), ak o p/14) = P(4).A a z Pu (T) = aix Pig (T) Replul T by t Pj (t) = 7 anklelt) P/+) - APIA) aue Calleol Chapman - Kolanogerov quation. Wholh care celso Called as forward & Beegwared equations.

. x in pulace of i in ich we put xsex Pxx Cat alt anexat toest) seonularly ② if we put izx, dj-x in Rg Páxht) = 7 and Pxx(+).

. x in pulace of i in ich we put xsex Pxx Cat alt anexat toest) seonularly ② if we put izx, dj-x in Rg Páxht) = 7 and Pxx(+).

! to waiting time de tebulion in casar chain with clasele clate spune ( Continuous lebre amas por relati x(): Contineaus maxor Preeus weth docieli State spoec. N-40, 12, 76, - - 1,--) trasition, ito] Po & X (THA) = 71 XCD=163 = Pj (1) - -0,1,2 .--,+20 Prod xu) -51X00)= i30 Pg (4) 02 Pg (t) 1 & 0,3 EP) ()=1 B (t) = Prd x H) = }} be the state probability cut epocht, BHI=PrYX H=1} Pod X(t) =3 & X00)=1} - m4810) = c] P[XH) -51%(0) =() - Pr {X10)=r] Pylt) EBW=1 ttco Que TPM PH)= (Pig (A)) suppen, a P.; 10) = 87, we get PCO) = I. @

Pos Continuary & afferentele 1 l differentiable for +70 The weating time for a change of st x 14) - honcogeneous mastlov pre brocis. x H) - XO) = ''Cannon) is rondom row able T to waiting time. pool ?>s++ / XC)=1] = Prof 2>S++ 1 x(0) =<,t>S] R$ 2751X002 103 - R4T>s+4 1x03) = 1? Pr *T>$/X10)=() FCH) = Proy r> + 1X60)=c) F (8+4) F (S). F(A) & 5,40 while 6 satisfied. off Elto y lhe farm F(A)=ént, tro, aco) Thus, the wenting time I fallous. an expenstiel dotre button will pacarmelina I have med deffail protation. But the problem is some.

Pos Continuary & afferentele 1 l differentiable for +70 The weating time for a change of st x 14) - honcogeneous mastlov pre brocis. x H) - XO) = ''Cannon) is rondom row able T to waiting time. pool ?>s++ / XC)=1] = Prof 2>S++ 1 x(0) =<,t>S] R$ 2751X002 103 - R4T>s+4 1x03) = 1? Pr *T>$/X10)=() FCH) = Proy r> + 1X60)=c) F (8+4) F (S). F(A) & 5,40 while 6 satisfied. off Elto y lhe farm F(A)=ént, tro, aco) Thus, the wenting time I fallous. an expenstiel dotre button will pacarmelina I have med deffail protation. But the problem is some.

Pos Continuary & afferentele 1 l differentiable for +70 The weating time for a change of st x 14) - honcogeneous mastlov pre brocis. x H) - XO) = ''Cannon) is rondom row able T to waiting time. pool ?>s++ / XC)=1] = Prof 2>S++ 1 x(0) =<,t>S] R$ 2751X002 103 - R4T>s+4 1x03) = 1? Pr *T>$/X10)=() FCH) = Proy r> + 1X60)=c) F (8+4) F (S). F(A) & 5,40 while 6 satisfied. off Elto y lhe farm F(A)=ént, tro, aco) Thus, the wenting time I fallous. an expenstiel dotre button will pacarmelina I have med deffail protation. But the problem is some.

is as be pure Birthe so a newssion process H2H) Perssion proess meon nt Pi,141 (st) = Pr & the proems gars to state it from state intime st] = Prof one enent oceur in tine st) = Pr JN(8t)=1] anst+o(at) Poi (at) = to an -1-ast+ocot) Piz (st) = o(st), tie it! ao, it on asiant, ay, 20 & 30, it! matux A=/ag) = 1-2 a orol Jaitl, 1+2 Peilt) = -ale, ilt) Pélt) = -2Poltta Pi, a ilt), 6 lt2 Pr of H 2t) =}} & 1010)=1, Po 1020, nto, BlH2Pg 14), 720. Bylt) = è at (at) J! Py (0)=1, jai, Pop (8) = 0, 143, Pg (t) 2 est cault (3-1)!

Peg (t) = enat cat si . (J-i)! put, is sok, Jak x-xtk P(x-4) alta é at. (at) (x-xefk)! =enat (@djk - Proced

ses as Sebey but minullerad 6 intearat. Probehelet will be 5/7 n=100 an 38100 /או' י ה

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