UNIVERSITY OF DUBLIN TRINITY COLLEGE

```UNIVERSITY OF DUBLIN
MA22S6
PA
PER
TRINITY COLLEGE
Faculty of Science
school of mathematics
SF Science
Sample Paper, 2010
Course 22S6
Dr. Michael Peardon
SA
MP
LE
Attempt THREE questions.
All questions carry 20 marks.
Marks for each part of the question are shown in square brackets.
MA22S6
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Page 2 of 3
1. (a) Write an expression for P (A|B), the conditional probability of event A occuring given event B
occurs in terms of the probability of both events occuring and the probability of event B.
[3]
(b) Three fair dice are rolled. Compute the probability the first die rolled one given exactly two dice
rolled one.
[7]
(c) In a game show, a contestant is shown three doors and asked to select one. Hidden behind one
door is a prize and the contestant wins the prize if it is behind their chosen door at the end of the
game. The contestant picks one of the three doors to start. The host then opens at random one
of the remaining two doors that does not contain the prize. Now the contestant is asked if they
want to change their mind and switch to the other, unopened door. Compute the probabilities of
winning if
i. the contestant sticks with their first choice.
[5]
ii. the contestant switches to the unopened door.
[5]
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2. Two random numbers X and Y have joint probability density given by

 6 (x + y 2 ) if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1
5
fXY (x, y) =

0
otherwise
(a) State a condition on the joint probability density functions that indicates X and Y are independent.
Are X and Y independent?
(b) Find
i. The probability density function of Y alone
MP
ii. The probability density function of Y given X &gt; 21 .
[3]
[7]
[10]
3. (a) Define what is meant by the sample mean and median of a sequence of n random numbers
{X1 , X2 , . . . Xn }.
[5]
(b) If these n samples are all independent and identically distributed with probability density function

 λe−λx
x≥0
fX (x) =

0
otherwise
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compute the expected values of the median and sample mean of the sequence.
[10]
(c) Describe the bubblesort algorithm. Estimate how many data-swap operations are required on
average to sort n numbers using this method.
[5]
Page 3 of 3
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MA22S6
4. (a) What are the defining properties of a Markov process?
[5]
(b) A simple climate model has the system in three states; χ1 =warming, χ2 =stable and χ3 =cooling.
The system makes random, memoryless transitions between these three states, with the probability
of a transition being made each year from state χj into χi given by the (i, j) element of the matrix




M =

4
5
1
10
1
5
1
5
4
5
1
5
0
1
10
3
5




Find the probability the system is in state χ1 in two years time given it is currently in state χ1 .[10]
(c) Show Pi , the probability of the system being in state χi on average is given by
P1 =
1
1
3
, P 2 = , P3 =
8
2
8
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[5]
c UNIVERSITY OF DUBLIN 2010
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