The screens used for a certain type of cell phone are manufactured by 3 companies,

A, B, and C. The proportions of screens supplied by A, B, and C are 0.5, 0.3, and

0.2, respectively, and their screens are defective with probabilities 0.01, 0.02, and 0.03,

respectively. Given that the screen on such a phone is defective, what is the probability

that Company A manufactured it?

9. (a) Show that if events A1 and A2 have the same prior probability P(A1) = P(A2),

A1 implies B, and A2 implies B, then A1 and A2 have the same posterior probability

P(A1|B) = P(A2|B) if it is observed that B occurred.

(b) Explain why (a) makes sense intuitively, and give a concrete example.

10. Fred is working on a major project. In planning the project, two milestones are set up,

with dates by which they should be accomplished. This serves as a way to track Fred's

progress. Let A1 be the event that Fred completes the first milestone on time, A2 be

the event that he completes the second milestone on time, and A3 be the event that he

completes the project on time.

Suppose that P(Aj+1|Aj )=0.8 but P(Aj+1|Ac

j )=0.3 for j = 1, 2, since if Fred falls

behind on his schedule it will be hard for him to get caught up. Also, assume that the

second milestone supersedes the first, in the sense that once we know whether he is

on time in completing the second milestone, it no longer matters what happened with

the first milestone. We can express this by saying that A1 and A3 are conditionally

independent given A2 and they're also conditionally independent given Ac

2.

(a) Find the probability that Fred will finish the project on time, given that he completes

the first milestone on time. Also find the probability that Fred will finish the project on

time, given that he is late for the first milestone.

(b) Suppose that P(A1)=0.75. Find the probability that Fred will finish the project

on time

(4) Is the Markov chain whose transition matrix whose transition matrix is 0 0.4 0.67 regular? (Yes or No) (5) Is the Markov chain whose transition matrix whose transition matrix is 0.2 0 0.8 0.6 0.4 0 regular? (Yes or No)(9) Stokes' Theorem: Stokes' Theorem converts a circulation calculation into a surface calculation through the formula where F is a given vector field, Curl F = V x F. S is a surface enclosed by the closed curve C, and 7 is the outward pointing unit normal. (a) Find the circulation of the vector field F = yi - r } on the circle Cox' + y' = 9, = =0 directly. (b) Find the circulation of the vector field in part (a), by converting the calculation to a surface calculation using Stoke's theorem, on the surface S :x+y' + 2 =9,2 20.1. (a) Define the Markov property of a discrete Markov chain. [5 marks] (b) Explain the difference between a homogeneous and inhomoge- neous Markov chain. [5 marks] (c) A three-state Markov chain has transition probability matrix, P, given below 1 2 3 1/ 0.8 0.1 0.1\\ P -2 0.15 0.75 0.1 3 0.06 0.04 0.9 If for all large n /0.3 0.2 0.5 0.3 0.2 0.5 0.3 0.2 0.5 what is the long-term proportion of time the chain spends in each of the three states? [5 marks) (d) State, with proof, whether the Markov chain in the previous ques- tion is reversible. [5 marks] (e) Assume that rows 1, 2 and 3 in the transition probability matrix, P correspond to transitions from states r, s and q, respectively of the chain. If the chain is in state q at some point i, what is the probability it was in state r at point i - 2? [5 marks]