数学马尔科夫链代写 Math 632代写 Exam代写 数学代写

Math 632 – Midterm Exam

数学马尔科夫链代写 • 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

• 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: P_x(A) is the probability of the event A when the Markov chain starts in state x, P_µ(A) the probability when the initial state is random with distribution µ. T_y = min{n ≥ 1 : X_n = y} is the first time after 0 that the chain visits state y . ρ_x,y = P_x(T_y < ∞) . N_y  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 X_t 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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