March 27, 2015 Math 303 Assignment 9: Due Wednesday, April 8 at start of class Note: Benjamin Wallace’s MLC hours on Thursday April 9 are cancelled and replaced by Tuesday April 7, 12:30–3:30pm. I. Problems to be handed in: 1. Particles are emitted by a radioactive substance according to a Poisson process of rate λ. Each particle exists for an exponentially distributed time, of rate µ, independent of the other particles, before disappearing. Let X(t) be the number of particles existing at time t. This is a birth and death process. (a) Determine the birth and death rates. (b) Find the limiting probabilities Pj . 2. Customers arrive, at rate λ, at a business with a single server. The service time has rate µ. Customers will wait in line if there is at most one other customer already waiting (in addition to one being served), but otherwise will go away. This defines a birth and death process. (a) What are the birth and death rates? (b) Determine the limiting probabilities. (c) Suppose λ = 1 and µ = 34 . What fraction of potential customers are lost, and what fraction of the time is the server idle? 3. Consider a birth and death process with states 0, 1, . . . , N , birth rates λn = αn(N − n), and death rates µn = βn2 (for 0 ≤ n ≤ N ). Find the limiting probabilities. 4. A factory consists of N identical machines, each operated by an employee. When a machine breaks down, its operator immediately begins to repair it. Each machine breaks down at rate µ, and each repair independently takes an exponential time of rate λ. Let X(t) denote the number of machines that are working at time t. This defines a birth and death process. (a) Determine the birth and death rates. (b) Determine the limiting probabilities. (It is a certain binomial distribution.) (c) Suppose that N = 50, λ = 10, µ = 1. What is the average number of machines that are operating, in the long run? 5. We model the changing relationships between 4 people as follows. Each pair of the group may or may not have a relationship, and if they do then we say the pair is linked. Suppose that any pair of unlinked individuals will become linked in a small time interval of length h with probability αh + o(h). Also, a linked pair becomes unlinked in a small time interval of length h with probability βh + o(h). The number X(t) of linked pairs at time t forms a birth and death chain. (a) Determine the birth and death rates. (b) Determine the limiting probabilities. 1 II. Recommended problems: These provide additional practice but are not to be handed in. The topics from the last weeks of class have few problems to be handed in for marking. It is highly recommended that the following problems on these topics be done before the final exam. Chapter 6 #2*, 4*, 7*, 11*, 16*, 19*, 24*, 28*. Quote of the week: It is seen in this essay that the theory of probabilities is at bottom only common sense reduced to calculus; it makes us appreciate with exactitude that which exact minds feel by a sort of instinct without being able ofttimes to give a reason for it. It leaves no arbitrariness in the choice of opinions and sides to be taken; and by its use can always be determined the most advantageous choice. Thereby it supplements most happily the ignorance and the weakness of the human mind. If we consider the analytical methods to which this theory has given birth; the truth of the principles which serve as a basis; the fine and delicate logic which their employment in the solution of problems requires; the establishments of public utility which rest upon it; the extension which it has received and which it can still receive by its application to the most important questions of natural philosophy and the moral science; if we consider again that, even in the things which cannot be submitted to calculus, it gives the surest hints which can guide us in our judgments, and that it teaches us to avoid the illusions which ofttimes confuse us, then we shall see that there is no science more worthy of our meditations, and that no more useful one could be incorporated in the system of public instruction. Pierre Simon Laplace, in A Philosophical Essay on Probabilities 2