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# 数学马尔科夫链代写 Math 632代写 Exam代写 数学代写

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## Math 632 – Midterm Exam

There are six problems on the exam, and they have multiple parts. The total number of points is 100.

Your solutions must be fully uploaded before 8:00 pm CDT (Wisconsin local time). Late submissions will not be accepted and receive 0 points. If there are technical difficulties, email your instructors immediately. Be mindful of the passage of time so that you do not miss the opportunity to scan and upload your work.

You are allowed one 2-sided 8×11 (A4 size) sheet of notes. No other materials or outside help is allowed. In particular, you cannot communicate with anyone else or look up solutions on the internet. 数学马尔科夫链代写

Simplify your answer if it is possible with reasonable e evaluated. ↵ort. In particular, geometric series must be evaluated.

Please present your solutions in a clear manner. Justify your steps. A numerical answer without explanation cannot get credit. Cross out the writing that you do not wish to be graded on.

General notation for Markov chains: Px(A) is the probability of the event A when the Markov chain starts in state x, Pµ(A) the probability when the initial state is random with distribution µ. Ty = min{n ≥ 1 : Xn = y} is the first time after 0 that the chain visits state y . ρx,y = Px(Ty < ∞) . Ny  is the number of visits to state y after time 0.

### 1. 数学马尔科夫链代写

Buses go by my bus stop such that the interarrival times are uniformly distributed in the interval [0, 1]. Suppose I come to the bus stop after the process has been going on for a long time. Calculate the expected amount of time I have to wait for a bus under the following two scenarios.

(a) As I arrive at the bus stop, I see that my previous bus just pulled out and is driving away.

(b) I have no information about the departure time of the previous bus.

### 2.

Toss a die five times. Let X denote the number of “3”s within the outcomes, and let Y be the number of appearance of odd numbers. Compute E[Y |X] and E[X2Y |X].

### 5.

Let N be a homogeneous rate λ Poisson process on (0, 1). Let 0 <a< 1.

(a) Conditional on N(0, 1) = 2, what is the probability that the two points in (0, 1) are within distance a of each other?

(b) What is the probability that the two first points of N are within distance a of each other without any conditioning?

### 6. 数学马尔科夫链代写

A barbershop has two barbers, and it has two chairs for service and one chair for waiting. The arrival rate of customers is λ and the service rate of each barber is µ. (The unit is # people per hour.) A customer who comes when both barbers are busy would decide to leave with probability 1/2 and to sit on the waiting chair with probability 1/2. A customer will be served as soon as one of the barbers is available. However, a waiting customer waits only for an exponentially distributed amount of time with mean 20 minutes and then leaves if she/he does not get served. 数学马尔科夫链代写

(a) Let Xt be the Markov chain that models the number of customers inside the barbershop at time t. Give the arrow diagram, routing matrix R, and generator matrix Q of this Markov chain.

(b) Find the long term fraction of time that both barbers are busy. (Give a formula in terms of λ and µ.)

(c) Suppose at time 0 both barbers are busy and the waiting chair is empty. What is the probability that both of them are busy and no one enters the barbershop throughout the time interval [0, h]?

(d) Suppose presently both barbers are busy and the waiting chair is empty. What is the probability that all of the next three customers will have to take the waiting chair?

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