At a Certain Stage, an Easily Communicable Disease has Spread to Infect 1 %: Principles of Applied Probability Assignment, SIM, Singapore

Assignment Questions:

Question 1

At a certain stage, an easily communicable disease has spread to infect 1 % of the population in a closed community. A fairly accurate diagnostic screening test is being developed for the disease, recording a positive result for 90 % of tested diseased individuals while giving a false result for 5 % of tested individuals who are without the disease.

(a) Compute the probability of a positive test result for an individual chosen at random from the population?

(b) Compute the probability of a positive tested individual actually having the disease?

Question 2

The probability generating function (p.g.f.) of a random variable, W, is given by

flw(s) = (‘)6 3 (1 + 2s)6

By writing flw(s) as a product of 2 independent p.g.f.’s, show that W can be expressed in terms of another random variable, X, which is binomial. Specify the exact distribution of X

Question 3

The incidence of earthquakes in a particular country may be modeled as a Poisson process occurring at an average rate of one every fifteen months.

(a) Compute the probability that over a period of ten years, there will be at least four earthquakes;

(b) Compute the probability that the waiting time between two consecutive earthquakes exceeds three years.

Question 4

Events in a Poisson process occur at a rate 2(t) = 3t

(a) Compute the expected number of events occurring by time t.

(b) If observation starts at time t = 0, calculate the probability P (T1 > t) for the waiting time Ti until the first event.

(c) Compute the expected waiting time until the first event, E [T1].

(d) If T is the time from the start of observation until the first event after time co-occurs, determine P (T > t)

[Hint: You may want to use the result:  L00 00 e-ct(D2 dC0 = .\11 1 a

Question 5

During shop opening hours, students visit a small campus shop according to a Poisson process at an average rate of one every 10 minutes. Independent of other students, each student makes Y purchases, with probability distribution:

Y 0 1 2 3 P(Y=y) 0.5 0.3 0.1 0.1

(a) Find the mean and variance of Y.

(b) Hence find the mean and variance of the total number of purchases made over a three-hour period.

Question 6

Let 11(s) = 1 — a (1 — sY 3 , where 0 < a < 1 and 0 < 13 < 1.

(a) Show that 11(s) is a proper probability generating function (p.g.f.) of the random variable Z, that is, it defines a proper probability distribution.

(b) Derive the p.g.f. fl, (s) , of the random variable Z., in the usual notation where fin(s) =  (c) Find expressions for P(Zii = 0), P(Zii = 1) and P(Zii = 2).

Question 7

In an unrestricted random walk starting at the origin, the ith step, Zi is such that P(Zi = 2) = p, P(Zi = -1) = 1- p

(a) Find the mean and variance of Zi. Hence write down the mean and variance of Xn, the position of the particle after n steps.

(b) Derive the probability distribution of Xn, showing all detailed workings.

(c) When p = 1/3, compute E [X20], var (X20), P(X20 = 0) and P(X20 = 1).

Question 8

In a market with 2 brands of bubble teas (Brands A and B), any Brand A customer will switch brand in the next purchase with probability 0.2, while any brand B customer will switch brand with probability 0.7.

We model the sequence of a customer’s brand purchase choice from one purchase to the next as a 2-state Markov Chain with state 0 (brand A) and state 1 (brand B).

(a) Write down P, the transition matrix of the Markov process, with the usual notations.

(b) Determine the market share of the 2 brands after they have been around the market for quite some time.