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University of Nottingham
School of Applied Mathematics
HG2MPN: Probabilistic & Numerical Techniques for Engineers
Assignment
(Coursework)
Hand in by 4.30 pm on 24 March 2014
Notes:
1. The following exercises are to be used for coursework assessment in the module HG2PMN. Credit
will be given for relevant working associated with solving the problems as well as the answers.
Results should be displayed to an accuracy of 4 decimal places.
2. An indication is given of the weighting of each section of a question be means of a figure enclosed by
square brackets, e.g. [20], immediately following that question.
Question 1
(a) There are 4 possible causes for an explosion at a construction site, namely static electricity (A),
malfunctioning of equipment (B), carelessness (C) and sabotage (D). It is estimated that an
explosion would occur with probability 0.35 due to A, 0.2 due to B, 0.5 due to C and 0.6 due to D.
It is also reported that the prior probabilities for the four causes of the explosion are 0.2 for A,
0.45 for B, 0.25 for C and 0.1 for D.
(i) Find the posterior probabilities.
(ii) What is the most likely cause of the explosion? Explain your answer. [10]
(b) The probability density function of the measurement of sulphur dioxide concentration (in
in a chemical compound (in ) is
( { (
(i) Find the normalization constant .
(ii) Find the cumulative density function (cdf) of .
(iii) The content of another pollutant is related to such that
(
). Find the
probability density function of .
(iv) Calculate the probability of falling in the range of . [15]
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Question 2
(a) Jobs are sent to a printer at an average of 4 jobs per hour. (i) What is the expected time between jobs?
(ii) What is the probability that a job is sent within 5 minutes? (iii) How long does it take at most, to have the arrival of the first job with a probability of 0.9?
Give your answer in minutes.
(iv) Suppose that the first job was sent at 9.30 a.m. * What is the probability that the printer gets the second job before 9.45 a.m.?
* What is the probability that the second job arrives after 9.45 a.m.? How do you relate
this to a Poisson distribution?
[15]
(b) An enhancement is performed in a type of plant in order to improve the yield of the plant. With such enhancement, the germination rate is claimed to be 90%. To evaluate, 60 seeds are planted
in a greenhouse and the number of seeds that germinate is recorded. (i) Explain why is a binomial random variable. (ii) What is the probability that at least 55 seeds will germinate? (iii) Find the mean and standard deviation of (iv) Repeat (ii) using normal approximation method. Determine the suitability of the normal
approximation. [15]
Question 3
In an easy dart shooting game, a dartboard with three concentric circles of radii
√ , 1 and √ metres is
used. Shots within the inner circle counts 5 points, in the next ring 3 points and between second and third
ring 2 points. Shots outside the board count zero. The distance of a shot from the centre of the board is a
random variable with the density function (
( , .
(a) Find the probabilities that a shot falls
(i) within the first (smallest) circle from the centre
(ii) between the first and the second circle
(iii) between the second and the third circle
(iv) outside the third (largest) circle [10]
(b) Suppose that the game costs $15 with 5 attempts of shots. The total score that a player gets is
the sum of scores obtained from the 5 shots, and the player will be paid $ to compliment his
success. Explain, with statistical evidence, whether this is a fair game. [10]