Exam 12: Stocastic Processes

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.

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.

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?

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.

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.

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.

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?

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.

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?

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.

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.

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.

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?