BU7527
Assessment
Mike Peardon — mjp@maths.tcd.ie
School of Mathematics, TCD
Michaelmas Term 2015-16
Attempt ALL questions. Submit your written solutions to the School of Business
by 5pm on Monday 16th November. Electronic submissions should be made
via the blackboard.tcd.ie web-page for the module. The total mark from
this assignment will count for 30% of your final grade for this module. Late
submissions will incur a penalty and may not be assessed.
Section A
1.
(a) An insurance company provides cover for fires in factories in a city. On average, there are eight fires a year distributed according to a Poisson process.
Find:
i. The probability there are no fires in a two-month period.
ii. The probability there are more than four fires in a six-month period.
iii. In one year, there are six fires. What is the probability they all occured
in the first three months of the year?
For simplicity, you may assume a year is twelve equal-length months.
(b) A die is rolled until two sixes have been seen in total but not necessarily in
consecutive rolls. Let the random number K be the total number of rolls.
Find:
i. P (K = k) for all possible values of k.
ii. P (K = k|A) with event A having occured when the first roll was a six.
(c) A finance company has a model to predict the rise or fall of a commodity
price at the end of the next month. Each month when the price falls, the
model successfully predicts the fall with probability 60%. In a month where
the price rises, the model falsely predicts a fall with probability 10%. On
average, the price of the commodity rises 10 months in a year. This month,
the model predicts a fall. What is the probability the price will in fact fall?
1
2. Two random numbers X and Y take values in the ranges X ∈ [0, 2] and Y ∈ [0, 1]
respectively and are drawn from the joint probability density function
fX,Y (x, y) =
3 2
x y +x
10
(a) Find the probability density functions of X and Y alone. Are X and Y
independent?
(b) Find the expected values of X and Y , their standard deviations and the
covariance, E[XY ] − E[X]E[Y ].
(c) Find P (Y < X).
3. Three broadband providers in a city, A, B and C run aggressive door-to-door advertising campains. Each month a customer of company A switches to provider
B with probability 2% and has a 3% chance of switching to provider C. A customer of provider B has a 8% chance of switching to C and a 1% chance of
switching to A. A customer of provider C has a probability of 2% of switching to
provider A and a 3% chance of switching to B. There are 10,000 customers in
the city in total. Find the long-time average expected number of customers for
each company.
4. Using a spreadsheet and 100 samples, estimate the integral
I=
Z 1
dx
−1
e−0.1x
x3 + 2
making sure to quote a 95% confidence interval for your result. Improve your
estimate using a control variate, f˜(x) = x31+2 , given
Z 1
−1
dx
x3
1
= 1.04154
+2
2
Section B
1. The price change Xt , with X0 = 0 of a commodity at time t is modelled as a
random walk with the rule that when Xt > 0,
(
Xt + 1 with probability 0.7
Xt − 1 with probability 0.3,
(
Xt + 1 with probability 0.5
Xt − 1 with probability 0.5,
Xt+1 =
and
Xt+1 =
otherwise. Use a spreadsheet package to simulate this system up to t = 15,
sampling the path at least 100 times and estimate by Monte Carlo simulation:
(a) The probabilities P (X15 = k) for all allowed values of k.
(b) E[X15 ]
2. Consider a dice game in which you are given at most n rolls to get the highest
score possible. Your final score is the number showing on the last dice rolled
when you either run out of turns or you choose to stop. Find the optimal strategy
for deciding when to stop rolling and take the current dice score. Compute the
expected value of your score when this strategy is followed for all values of n.
3. A flea sits in a square of a 5 × 5 square chess-board. Each second, the flea jumps
at random to one of the neighbouring squares on the board, with all allowed
possibilities (there are 2, 3 or 4 neighbours) equally likely to be chosen. The flea
never jumps off the chessboard. Find the long-term average probability the flea
is on a square at the edge of the board.
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