Janacek statistics tables (2009 edition) are available on your desk. Do not turn over until you are told to do so by the Invigilator.

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination CALCULUS AND PROBABILITY MTHB4006Y Time allowed: 2 Hours Attempt THREE questions. Janacek statistics tables (2009 edition) are available on your desk. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHB4006Y Module Contact: Dr Peter Milne, MTH Copyright of the University of East Anglia Version: 1

2 (i) Explain carefully what is meant by any three of the following: (a) the probability mass function of a discrete random variable; (b) a random experiment; (c) sample space; (d) events A and B are disjoint; (e) events A and B are stochastically independent. (ii) Let S be a sample space and let A and B be events in S. Show that [ ] (a) P (A B c ) (A c B) = P (A B) P (A B); (b) P (A) = P (A B)P (B) + P (A B c )P (B c ). [8 marks] (iii) A professor of mathematics goes to a conference and wishes to attend a particular session. The probability that he will go to the wrong room is 0.8, the probability that he will arrive at the wrong time is 0.6 and the probability that he will arrive at the wrong room and at the wrong time is 0.5. Calculate the probability that (a) he will arrive either at the wrong room or at the wrong time; (b) he will arrive at the wrong room, given that he arrives at the right time; (c) he will arrive at the right time, given that he arrives at the wrong room.

3 (i) A continuous random variable X has probability density function f. State the definition of (a) the expectation E(X) of X ; (b) the variance V (X) of X. (ii) Define the continuous random variable T to be the waiting time between successive events in a Poisson process of intensity α. Show that (a) the probability density function of T is { 0 if t < 0 f(t) = αe αt if t 0; (b) E(T ) = 1 α ; (c) V (T ) = 1 α. 2 [10 marks] (iii) Lemmings jump off a cliff according to a Poisson process of intensity 6 per minute. A naturalist waits at the foot of the cliff and counts the number of lemmings arriving in a two minute period. (a) What is the probability that at most 12 lemmings arrive? (b) What is the probability that at least 4 lemmings arrive? (c) What is the expected waiting time between the first and fourth lemmings? MTHB4006Y PLEASE TURN OVER Version: 1

4 (i) At Fred Scruntley s Automotive Works, 30% of complaints are about defective spigots. On a certain day 16 customers make complaints. (a) What is the probability that at most 7 complaints are about defective spigots? (b) What is the probability that at least 6 complaints are about defective spigots? (c) If it is known that at most nine complaints were about defective spigots, what is the probability that at most 11 were about something else? [10 marks] (ii) The length of time for which a unit continues to function is a continuous random variable L. The reliability function R(t) of the unit is defined to be the probability that it still functions at time t: R(t) = P (L > t). The expected life of the unit is given by E(L) = 0 R(t)dt. Four units are connected to form the system shown below. (a) If the four units have reliability functions R 1 (t),..., R 4 (t), show that the reliability function of the system is R(t) = R 1 (t)r 2 (t) + R 3 (t)r 4 (t) R 1 (t)r 2 (t)r 3 (t)r 4 (t). (b) If the four units are all identical with reliability function R i (t) = e 3t (i = 1, 2, 3, 4), find the expected life of the system. [10 marks]

5 A mouse moves through the maze shown, changing room at time t = 0, 1, 2,..., by choosing any exit at random from the room currently occupied. Rooms 1 and 4 both contain a large piece of cheese; once the mouse enters either of these rooms it remains there for the duration of the experiment. This system can be described by a Markov chain with states {1, 2, 3, 4}, where the chain is in state i if the mouse is in room i. (i) Write down the transition matrix for this chain. [5 marks] (ii) For i = 1, 2, 3, 4, find the probability that the mouse is in room i after two timesteps, given that its initial room is chosen at random. (iii) (a) Find the fundamental matrix of this Markov chain. (b) Let π ij be the probability that the process is absorbed in state j given that it starts in state i, and let Π be the 2 2 matrix whose elements are π ij, i = 2, 3, j = 1, 4. Find Π. [7 marks] (iv) Find the expected number of timesteps elapsing before the mouse reaches the cheese when it starts in (a) room 2; (b) room 3. END OF PAPER