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# 随机过程课业代做 526 Stochastic Processes代写 作业代写

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## Homework 3

The maximum number of points you can receive for this homework is 24.

### 1. (4 pts)

Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 20 minutes to clean the kitchen. Kelly takes an exponentially distributed amount of time with mean 30 minutes to clean the bathroom. The first one to complete their task

will go outside and start raking leaves, a task that takes an exponentially distributed amount of time with a mean of one hour. When the second person is done inside, she will help the other, and raking will be done at rate 2 per hour. (Of course the other person may already be done raking in which case the chores are done.) What is the expected time until the chores are all done?

### 2. (4 pts) 随机过程课业代做

Two types of consultations occur at a data base according to two independent Poisson processes: read consultations arrive at the rate λR and write consultations arrive at the rate λW .

(a) What is the probability that the time interval between two consecutive read consultations is larger than t > 0?

(b) What is the probability that during the time interval [0, t], at most three write consultations arrive?

(c) What is the probability that the next arriving consultation is a read consultation?

(d) Determine the distribution of the number of arrived read consultations during [0, t], given that in this interval a total number of n consultations occurred.

### 3. (4 pts) 随机过程课业代做

Customers arrive at a full-service gas station, with two pumps, at rate of 30 cars per hour. However, customers will go to another station if there are at least four cars in the station: i.e., two being served and two waiting. Suppose that the service time for customers is exponential with mean 5 minutes. Denote by Xt the number of cars at the gas station at time t (both waiting and being served).

a) In the long-run, how many cars per hour are served at the station?

b) If the gas station is currently empty, what is the expected time until it becomes full?

### 4. (4 pts) 随机过程课业代做

Cars pass a certain street location with identical speeds, according to a Poisson process with rate λ > 0. A person at that location needs T units of time to cross the street, so waits until it appears that no car will cross that location within the next T time units.

(a) Find the probability that the waiting time is 0.

(b) Find the expected waiting time.

(c) Find the total average time it takes to cross the street.

(d) Assume that, due to other factors, the crossing time in the absence of cars is an independent exponentially distributed random variable with parameter µ > 0. Find the total average time it takes to cross the street in this case.

### 5. (3pts) 随机过程课业代做

Let X be a random variable with values in {0, 1} and let Y be a random variable with values in {0, 1, 2}. Initially we have the following partial information about their joint probability mass function:

Subsequently we are provided with the following information:

(i) Given X = 1, Y is uniformly distributed.

(ii) pX|Y (0|0) = 2/3.

(iii) E[Y |X = 0] = 4/5.

Use this additional information to fill in the missing values of the joint probability mass function table. Provide computations for your answers.

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