stochastic process
STAT4207/5207 Homework assignment 4 Due by April 15, 2021. Write down the complete derivations for full credits. Question 3 and 4 are bonus questions (worth 12 and 13 points respectively). Exercise 1 (55 points) For a standard Brownian motion (Bt : t ≥ 0) a) Find P(B2 −B1 > 1|B0.5 = 2). b) Consider the process Xt = Bt − tB1 0 ≤ t ≤ 1. i) Derive E[Xt] and Cov(Xt, Xs) for some fixed t, s ∈ [0, 1] and specify the joint distribution of Xt and Xs. ii) Calculate P(X0.15 > 0.2). iii) Find E[Xs|Xt = 1] for some fixed s, t ∈ [0, 1] such that 0 < s="">< t.="" c)="" consider="" the="" process="" xt="µt+" σbt="" t="" ≥="" 0.="" for="" constants="" µ="" ∈="" r,="" σ=""> 0. Derive E[Xt] and Cov(Xt, Xs) for some fixed t, s ∈ [0, 1] and specify the joint distribution of Xt and Xs. d) Consider the process Xt = e µt+σBt t ≥ 0. for constants µ ∈ R, σ > 0. i) Prove that, for Z ∼ N (0, 1), E[eλZ ] = eλ 2/2 ∀λ ∈ R. ii) Use your previous answer and the self-similarity property of Brownian motion to show that EXt = e(µ+0.5σ 2)t. iii) Derive Cov(Xt, Xs) for some fixed t, s ∈ [0, 1]. iv) Give the distribution of Xt for some fixed t > 0. e) Find the distribution of the random variable (Bt −Bs)2 (Bv −Bt)2 for 0 < s="">< t="">< v. exercise 2 (15 points) suppose a factory has two machines that are maintained by a single technician. furthermore, suppose that machine i = 1, 2 functions for an exponential time with rate µi before it breaks down. the repair time for each of the machines is exponentially distributed with rate α. the repair and lifetimes of the two machines are completely independent. model this system as a continuous-time markov chain. can it be modeled as a birth and death process? explain. 1 exercise 3 (bonus question) consider a continuous time markov chain with states {0, 1, 2, 3} and generator q that is given by q = −6 3 3 0 2 −4 0 2 2 0 −5 3 0 2 1 −3 a) let a = {1, 3}. find e[τ0,a]. b) find the probability that, starting in state 0, state 1 is visited before state 3. exercise 4 (bonus question) suppose each time a machine is repaired it remains up for an exponentially distributed amount of time with parameter λ. it then fails, and its failure is either of two types. if it is a type 1 failure, the time to repair is exponentially distributed with parameter µ1. if it is type 2 failure, the repair time is exponentially distributed with parameter µ2. assume that a failure is, independent of the time it took for the machine to fail, a type 1 failure with probability p and a type 2 failure with probability 1− p. a) what is the proportion of time that the machine is down due to a type 1 failure? b) what is the proportion of time that the machine is down due to a type 2 failure? c) what proportion of time is the machine working? 2 v.="" exercise="" 2="" (15="" points)="" suppose="" a="" factory="" has="" two="" machines="" that="" are="" maintained="" by="" a="" single="" technician.="" furthermore,="" suppose="" that="" machine="" i="1," 2="" functions="" for="" an="" exponential="" time="" with="" rate="" µi="" before="" it="" breaks="" down.="" the="" repair="" time="" for="" each="" of="" the="" machines="" is="" exponentially="" distributed="" with="" rate="" α.="" the="" repair="" and="" lifetimes="" of="" the="" two="" machines="" are="" completely="" independent.="" model="" this="" system="" as="" a="" continuous-time="" markov="" chain.="" can="" it="" be="" modeled="" as="" a="" birth="" and="" death="" process?="" explain.="" 1="" exercise="" 3="" (bonus="" question)="" consider="" a="" continuous="" time="" markov="" chain="" with="" states="" {0,="" 1,="" 2,="" 3}="" and="" generator="" q="" that="" is="" given="" by="" q="" −6="" 3="" 3="" 0="" 2="" −4="" 0="" 2="" 2="" 0="" −5="" 3="" 0="" 2="" 1="" −3="" ="" a)="" let="" a="{1," 3}.="" find="" e[τ0,a].="" b)="" find="" the="" probability="" that,="" starting="" in="" state="" 0,="" state="" 1="" is="" visited="" before="" state="" 3.="" exercise="" 4="" (bonus="" question)="" suppose="" each="" time="" a="" machine="" is="" repaired="" it="" remains="" up="" for="" an="" exponentially="" distributed="" amount="" of="" time="" with="" parameter="" λ.="" it="" then="" fails,="" and="" its="" failure="" is="" either="" of="" two="" types.="" if="" it="" is="" a="" type="" 1="" failure,="" the="" time="" to="" repair="" is="" exponentially="" distributed="" with="" parameter="" µ1.="" if="" it="" is="" type="" 2="" failure,="" the="" repair="" time="" is="" exponentially="" distributed="" with="" parameter="" µ2.="" assume="" that="" a="" failure="" is,="" independent="" of="" the="" time="" it="" took="" for="" the="" machine="" to="" fail,="" a="" type="" 1="" failure="" with="" probability="" p="" and="" a="" type="" 2="" failure="" with="" probability="" 1−="" p.="" a)="" what="" is="" the="" proportion="" of="" time="" that="" the="" machine="" is="" down="" due="" to="" a="" type="" 1="" failure?="" b)="" what="" is="" the="" proportion="" of="" time="" that="" the="" machine="" is="" down="" due="" to="" a="" type="" 2="" failure?="" c)="" what="" proportion="" of="" time="" is="" the="" machine="" working?="">