Deck 12: Stocastic Processes

Question 1

A custom auto manufacturer runs 24 hours per day and produces automobiles according to a Poisson process with rate λ. If over a 2-day period exactly 6 autos are produced, what is the probability that an auto was produced every 8 hours for those 2 days?

Answer 1

6!/(6^6)

Question 2

Construct a transition probability matrix of a Markov chain with state space {1,2,…,7,8} in which {1,2,3} is recurrent having period 2, {4,5,6,7} is recurrent having period 3, and {8} is aperiodic transient.

Answer 2

$$\left( \begin{array} { l l l l l l l l } 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & .5 & 0 & 0 & .5 & 0 \\0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\end{array} \right)$$

Question 3

In a given town, on a given day, citizens can either go to the library (activity 1), go to the zoo (activity 2), or stay home (activity 3). Let Xₙ=i if on the nᵗʰ day of the year, the Shuzoku family does activity i. Suppose {Xₙ | n=1,…} is a Markov chain with transition probability matrix $$\left( \begin{array} { l l l } .23 & .21 & .56 \\.15 & .41 & .44 \\.22 & .08 & .70\end{array} \right) .$$ We know the Shuzoku family went to the zoo on day 1. What is the probability they did not go to the zoo the next two days?

Answer 3

.5827

Question 4

Let {Xₙ | n=1,…} be a Markov process with state space {1,2,3} and transition matrix $$\left( \begin{array} { l l l } .18 & .26 & .56 \\.11 & .61 & .28 \\.22 & .34 & .44\end{array} \right)$$ If X₁ is in state 1, find the expected number of transitions until getting to 2 or 3 for the first time.

Question 5

To practice as a real estate agent in Tallahassee, one must get a license and keep it valid. Your license expires based on how many homes you sell. Suppose that, for an agent with a new license, the time for the license to expire is an exponential random variable with mean 1/α and the time required to renew his or her license once expired is an exponential random variable with mean 1/β. Suppose that agents let their licenses expire and then renew it. If there are 12 agents at a certain agency, find the long-run proportion of time that only 2 of those agents have valid licenses.

Question 6

Geoff, Berniece, and Luis play a three-player game. Each time they play Geoff wins with probability .45, Berniece wins with probability .32 and Luis wins with probability .23. After each play the winner gets to wear a crown, which then goes to the winner of the next game. Find the long-run proportion of the time each player gets to wear the crown.

Question 7

Calls come in for a radio contest at a Poisson rate of 12 per minute. If 40 calls come in between 2:00pm and 2:04pm, what is the probability that only the last two calls arrived in the last 10 seconds of 2:03pm?

Question 8

At a croissant factory, sometimes the croissants come out too dark. After a dark croissant is produced, out of the next 9 croissants, exactly 1 is too dark. After each normal croissant (that is, a croissant with beautiful golden color), out of the next 14 croissants, exactly 12 of them are normal. Find the long run fraction of normal croissants produced.

Question 9

Each day, Quintana watches exactly one movie. She only watches action movies (genre 1), comedies (genre 2), and dramas (genre 3). Let Xᵢ, i=1,… be a sequence of random variables with Xᵢ=n if Quintana watches a movie of genre n on day i. Suppose that {Xᵢ} is a Markov process with probability transition matrix $$\left( \begin{array} { c c c } .2 & .27 & .53 \\.18 & .33 & .59 \\.22 & .1 & .67\end{array} \right)$$ Find the long run proportion of days she watches each genre.

Question 10

Consider the following probability transition matrix: $$\left( \begin{array} { c c c c c c } 0 & 1 & 0 & 0 & 0 & 0 \\.4 & .6 & 0 & 0 & 0 & 0 \\.3 & 0 & .4 & .2 & .1 & 0 \\0 & 0 & 0 & .3 & .7 & 0 \\0 & 0 & 0 & .5 & 0 & .5 \\0 & 0 & 0 & .3 & 0 & .7\end{array} \right)$$ Find the classes of this Markov chain and determine whether each is recurrent or transient.

Question 11

A certain auto shop is an M/M/1 queuing process in which customer arrivals follow a Poisson process with rate λ=3 customers per hour, and service times are exponential random variables with mean μ=1/4 hours per customer. Find the proportion of the time there are at most 2 customers waiting in the queue.

Question 12

Suppose that a bank knows that customers arrive at a Poisson rate of 8 per hour and the service time of a customer is exponential with mean 1/6 hours. The bank wants to minimize the number of idle tellers. If the system is an M/M/6 queuing system (i.e., it has 6 tellers), find the probability that there is an at least one idle server at any given time.

Question 13

Consider a Brownian motion {X(t) : t≥0} with variance parameter σ²=2. For what value of M can we be 90% sure that the motion does not surpass M in 6 seconds?