Question

( Markov jump process and Kolmogorov forward equation ) In a particular company,the salary scale has only two different levels. On average, an employee spends 2 years at level 1 before moving on to the higher level, or leaving the company. An employee at the maximum level spends an average of 5 years before leaving. Nobody is demoted, and promotion can occur at any time. Upon leaving level 1, the probability that an employee moves to level 2 is 50%.

(a) Explain how you could model this as a Markov jump process, commenting on any assumptions that you make. Draw the state diagram of the Markov process.

(b) From the average waiting time, find the diagonal element of the intensity matrix,and the transition probability matrix of the jump Markov process.

c) Show that the intensity matrix of the Markov jump process is given by

Q=\left(\begin{array}{ccc} -0.50 & 0.25 & 0.25 \\ 0 & -0.20 & 0.20 \\ 0 & 0 & 0 \end{array}\right)

(d) Use the Kolmogorov's forward equation to show that the probability of moving from level 1 to level 2 in t periods of time is given by

P_{12}(t)=\frac{5}{6}\left(e^{-0.2 t}-e^{-0.5 t}\right)

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Question 45696

(1 point) A biotechnology company produced 182 doses of somatropin, including 12 which were defective. Quality control test 14 samples at random, and rejects the batch if any of the random samples are found defective. What is the probability that the batch gets rejected?

Question 45695

(1 point) Assume that the monthly worldwide average number of airplaine crashes of commercial airlines is 2.2. What is the probability that there will be
(a) at most 5 such accidents in the next month?
(b) less than 4 such accidents in the next 3 months?
(c) exactly 8 such accidents in the next 5 months?

Question 45694

(1 point) A certain typing agency employs two typists. The average number of errors per article is 1 when typed by the first typist and 3.2 when typed by the second. If your article is equally likely to be typed by either typist, find the probability that it will have no errors.

Question 45693

(1 point) The mean number of patients admitted per day to the emergency room of a small hospital is 2. If, on any given day, there are only 7 beds available for new patients,what is the probability that the hospital will not have enough beds to accommodate its newly admitted patients?

Question 45692

nt) Given that x is a random variable having a Poisson distribution, compute the following:
\text { (a) } P(x=7) \text { when } \mu=0.5
P(x \leq 6) \text { when } \mu=4.5
\text { c) } P(x>3) \text { when } \mu=5.5
P(x<9) \text { when } \mu=4.5

Question 45691

B. Compute the probability that in the next 8 hours, more than 24 cars will arrive.
A. Determine the probability that over the next hour, only one car will arrive.
Cars arriving for gasoline at a Shell station follow a Poisson distribution with a mean of 9 per hour.

Question 45690

Complaints about an Internet brokerage firm occur at a rate of 5 per day. The number of complaints appears to be Poisson distributed.
A. Find the probability that the firm receives 4 or more complaints in a day.
Probability =%3D
B. Find the probability that the firm receives 22 or more complaints in a 5-day period.
Probability =

Question 45689

Flaws in a carpet tend to occur randomly and independently at a rate of one every 150 square feet. What is the probability that a carpet that is 9 feet by 13 feet contains noflaws?
Probability:

Question 45688

The number of accidents that occur at a busy intersection is Poisson distributed with a mean of 3.4 per week. Find the probability of the following events.
.4 or more accidents occur in a week.
A. No accidents occur in one week.
One accident occurs today.

Question 45687

(1 point) An airline company is considering a new policy of booking as many as 271 persons on an airplane that can seat only 260. (Past studies have revealed that only 89%of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 271 persons. not enough seats will be available.