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Probability: Theory and Applications

problem 1: Suppose the arrival times of phone calls in a help centre follow a Poisson process with rate 20 per hour (so the inter-arrival times are independent exponential random variables). Management are interested in simulating the arrival process in order to decide whether to employ a new customers services representative.

(a) Use the internet or a text book to nd an ex of a congruential unit random number generator that is di fferent to the one given in the lecture notes and lab exercises.

(b) Use your random number generator to describe why random numbers generated on a computer are called `pseudo random numbers'.

(c) Use your random number generator to generate 3 unit random numbers. State what`seed' is used and describe how this can be used to repeat simulation experiments.

(d) Assuming the memoryless property and a starting time of 1pm, use your simulated unit random numbers above to generate the arrival times of the next 3 calls arriving at the centre.

problem 2: The lifetime T (in days) of an electrical component has reliability function given by: R(t) = e 0.01t for time t > 0. An electrical system consists of four such components. The system continues to function if at least one component is still `alive' and the system is repaired (by replacing all the components) when all components have expired.

(a) Find the probability density function (PDF) of T, the lifetime of a component in the system.

(b) Show that this is a valid PDF.

(c) Find the reliability function for the system.

(d) Find the probability that the system lasts for longer than 1 year (365 days).

(e) Simulate the lifetime of the system.

(f) Set the random number seed using set.seed(1). Then simulate 20 lifetimes for the system. find out the sample mean lifetime of the system using your 20 simulated values.

problem 3: A drainage system overflows into a river causing pollution when the rainfall falling in an hour exceeds 50 mm. During any given year, the maximum hourly rainfall follows an exponential distribution with mean 20 mm. Find the return period (in years) of the critical event that causes pollution in the river.

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