MAS113 Introduction to Probability and Statistics
Exercises
You will be told which problems to work on each week at the lecture each Monday, and you
should try them before your class each Wednesday. Every two weeks, starting in week 3,
you will also be asked to hand in solutions to two questions as homework. You will be told
what the homework questions are after the tutorials, and you should hand them in at the
lecture on the following Monday.
Hints for questions marked with a * are given at the end of this booklet.
1. In each of the following cases, describe a suitable sample space for the experiment and
identify the event indicated as a subset of this sample space. [Do not try to assign
probabilities.]
(a) Experiment: toss a coin three times.
Event: the number of heads is even.
(b) Experiment: count the number r of red tomatoes and the number y of yellow
tomatoes grown by a gardener.
Event: there are more yellow tomatoes than red ones.
(c) Experiment: measure the quantities a, b, c (in cm) of rainfall in a day at each of
three weather stations A, B and C.
Event: it is driest at station A.
2. For S = N and subset A of S, consider the set function
g(A) = min(A),
the smallest element of the set A, with g(∅) defined to be 0. Give a counter-example
to show that g is not a measure.
3. * For any three subsets A, B, C of a sample space S (not necessarily disjoint), prove
that for any measure m,
m(A ∪ B ∪ C) = m(A) + m(B) + m(C) − m(A ∩ B)
−m(A ∩ C) − m(B ∩ C) + m(A ∩ B ∩ C).
You may quote the result that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
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4. * Show that if m and n are two measures on a sample space S, and c ≥ 0, then
(a) m + n is a measure on S, where (m + n)(A) = m(A) + n(A),
(b) cm is a measure on S, where (cm)(A) = c × m(A).
5. * Given a sample space S with A ⊆ S, prove that P (Ā) = 1 − P (A). Justify each step
in your proof carefully.
6. In the Champions League semi-finals, teams are chosen at random to play each other.
Two teams out of the four are selected at random for the first semi-final, and the
remaining two teams play each other in the second semi-final. Assuming that all
random selections are equally likely, if two of the four teams are English, what is the
probability that they are not drawn to play each other? Show clearly how you have
derived your probability.
7. A bookmaker is offering odds on the outcome of the 2017/18 Ashes series between
England and Australia. They offer odds of 4/7 against Australia winning the series, 5/2
against England winning the series and 5/1 against a drawn series. By converting these
odds to probabilities, show that these odds do not correspond to a valid probability
measure (i.e. they do not satisfy the rules of probability). If the odds for England
and Australia winning the series are unchanged, what should the odds against a drawn
series be to give a valid probability measure?
8. In the game show “Who wants to be a Millionaire?”, a contestant must answer 12
questions correctly to win £1,000,000. Each question is multiple choice, with four
possible answers. For each question, explain your reasoning carefully:
(a) If a contestant guesses each time, picking one of the four answers at random, what
is the probability of winning the million pound prize?
(b) Now suppose the contestant uses the three “lifelines”: for one question, two false
answers are eliminated; for another question, the audience votes for the correct answer; for a third question, the contestant may phone a friend for advice. Suppose
that both the audience and the friend give the correct answer on their respective
questions. The contestant uses the answers from the audience and the friend,
but otherwise picks an answer at random each time. What is the probability of
winning the million pound prize?
9. An online banking website requires its account holders to choose a four digit pin number, and a case-sensitive password made up of digits and letters (the letters can be
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upper or lower case). Suppose you choose your pin number and a six character password at random (for the password, each character has the same chance of being any
digit or upper or lower case letter). When logging in, the website asks you to specify
three digits from your pin number, and three characters from your password, all chosen
randomly, and will give you three attempts if you make a mistake. If someone tries to
log into your account, what is the probability that they would gain entry within three
attempts? Justify your answer.
10. In the card game of Blackjack, players attempt to get a total of 21 points in their hand.
An ace may be counted as 1 point or 11 points, and face cards (kings, queens, and
jacks) are counted as 10 points. For each question, explain your reasoning carefully:
(a) A player is dealt two cards from a standard deck of 52. What is the probability
that these two cards sum to 21 points?
(b) A player is dealt a 10 and a 6 for his hand, again from a standard deck of 52. His
opponent (the dealer) has drawn a 4. If the player draws one more card, what is
the probability that the player’s total will exceed 21 points? (You should assume
that an Ace will be counted as 1 point in this context.)
(c) Suppose instead two standard decks of cards are shuffled together, so that there
are 104 cards in total. A player is again dealt two cards. What is the probability
that these two cards sum to 21 points?
11. In the game show “Deal or no Deal?”, a contestant must first choose 5 boxes out of 22
to be opened. Each box contains a different amount of money. Once a box has been
opened, its contents can no longer be won. For each question, explain your reasoning
carefully:
(a) What is the probability that the contestant chooses the most valuable box?
(b) What is the probability that the contestant chooses at least one of the top 5 most
valuable boxes?
12. Two events A and B have P (A) = 0.4, P (B) = 0.3 and P (A ∪ B) = 0.58. Are A and
B independent?
13. In a group of 100 men, 55 are clean-shaven, 30 have beards and moustaches, 10 have
moustaches only, and 5 have beards only. One man is selected at random. For each
question, explain your reasoning carefully:
(a) What is the probability that the man has a beard?
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(b) If it is known that the selected man has a beard, what is the probability that he
has a moustache?
(c) In this group of 100 men, are the events of having a beard and having a moustache
independent for a randomly selected man?
14. Suppose England are going to play Spain at football in one month’s time. You judge
that if England’s star player is fit, your probability that England win is 0.3, but if he
is injured, your probability that England win is 0.2. If your probability that England’s
star player will be injured at the time of the game is 0.05, what is your probability that
England will win? If, after the match, you found out only that England won, what
would your probability be that the star player was injured? Justify your answers.
15. In a production line, it is estimated that 1 in every 200 items will be faulty. At the
quality control stage, each item is visually inspected, and it is believed that there is
a 90% chance of detecting a faulty item, and a 5% chance of mistakenly declaring a
non-faulty item as being faulty. If, at the inspection, an item is declared as faulty, what
is the probability that it genuinely is faulty? Define any notation that you introduce,
and justify your answer.
16. * An art dealer receives a shipment of five old paintings from abroad, and, on the
basis of past experience, she feels that the probabilities are, respectively, 0.76, 0.09,
0.02, 0.01, 0.02 and 0.10 that 0,1,2,3,4 or all 5 of them are forgeries. Since the cost of
authentication is fairly high, she decides to select one of the five paintings at random
and send it away for authentication. If it turns out that this painting is a forgery, what
probability should she now assign to the possibility that all the other paintings are also
forgeries? Note that the answer is not 0.1.
17. * Three prisoners, A, B and C, have been told by their jailer that one of them, chosen
at random, will be executed, and the other two will be freed. Prisoner A says to the
jailer,“I know that one of the other prisoners must be freed, so why don’t you tell me
which of the two it will be, or tell me one name at random if they are both to be freed?
It can’t change my chances of being executed.” The jailer replies, “Once I’ve told you
a name, that only leaves two people, so your chances of being executed will have gone
up to 1 in 2.” Who is right?
18. A man is on trial for the murder of his wife. The defendant is known to have been
violent towards his wife on at least one occasion. The defence argue that this evidence
is of little relevance, since fewer than 1 in 2500 men who are violent to their partners
go on to murder them. Assuming the figure is right, is the defence’s argument valid?
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19. An email spam filter applies the following test to each incoming email: if all of the
following conditions are true, the email is declared spam:
• the email has not been sent from a university account;
• the email has not been sent from a distribution list, to which the user is subscribed;
• the receiver has never sent an email to the sender;
• the email text does not contain the receiver’s name.
Suppose 80% of all emails are spam, 95% of spam emails are declared spam by the
filter test, and 5% of genuine emails are declared spam by the filter test. Calculate
the probability that an incoming email will be declared spam, and the probability of
a ‘false positive’: the probability that an email that is declared to be spam is in fact
genuine. Define any notation that you introduce, and justify your answer.
20. Form a list of 9 digits using the 9 digits in your student registration number and add
the digit 5 to the list. Let X be the number obtained by selecting at random one of
the 10 digits from the list, such that each of the 10 digits has the same probability
of being selected. Tabulate the probability mass function and cumulative distribution
function of X, for integer values of X from 0 to 10 inclusive. Sketch a plot of the
cumulative distribution of X, with the x-axis ranging from -1 to 10. Calculate the
standard deviation of X.
21. Suppose, for a lottery scratchcard, probabilities of different prizes are as follows
prize probability
£0
0.689
£2
0.300
£10
0.010
£100
0.001
If a scratchcard costs £1, and you buy one card, tabulate the probability mass function
and cumulative distribution function of your profit. Calculated the expectation and
standard deviation of your profit. Define any notation that you introduce.
22. A discrete random variable X has expectation 3 and variance 10. Let Y = (X + 1)2 .
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Explain what is wrong with the following, and derive the correct expectation of Y .
E(Y ) = E{(X + 1)2 } =
=
=
=
E(X 2 + 2X + 1)
E(X)2 + 2E(X) + 1
32 + 2 × 3 + 1
16.
(1)
(2)
(3)
(4)
23. On a European roulette wheel, the ball can land on one of the integers 0 to 36. Assume
that the ball is equally likely to land anywhere. Let X be a random variable that
takes the value 1 if the ball lands on an odd number, and 0 otherwise. Calculate the
expectation and variance of 5X.
Now suppose that the ball is spun 5 times, with Y the total number of times that the
ball lands on any odd number from 1 to 35. Does Y have the same expectation and
variance as 5X? Justify your answer.
24. * You are asked to provide your subjective probability that it will rain tomorrow. If
you provide a value q, you will be paid £q pounds if it rains tomorrow, and £(1 − q)
if it does not rain tomorrow. If your subjective probability of rain tomorrow is p, and
you want to maximise your expected earnings, should you be truthful and state q = p,
or should you state some other value for q? Your value of q must lie in the interval
[0, 1].
25. In a multiple choice test, there are 10 questions, with four answers per question. For
each question, only one out of the four answers is correct. If you were to pick one
answer at random for each question, calculate the probability of getting exactly 6 out
of 10 answers correct. Define any notation that you introduce, and justify your answer
26. A TV chef claims his free-range chickens taste better than battery-farmed chickens. 10
people are given a sample of each to taste, without knowing which is which, and are
asked to state which sample they prefer.
It is suggested that the participants cannot actually taste the difference, and are effectively choosing which sample they prefer at random. If this is true,
(a) what probability distribution would you use to describe the number of people who
say they prefer the free-range chicken?
(b) Using your distribution in (a), calculate the probability that the number of people
who say they prefer the free-range chicken is
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i. not 5;
ii. no more than 8.
Define any notation that you introduce, and justify your answers.
27. * Let X1 ∼ Bin(n1 , p1 ) and X2 ∼ Bin(n2 , p2 ), and assume that X1 and X2 are independent.
(a) Find the mean and variance of X1 + X2 .
(b) If p1 = p2 , what distribution would you expect X1 + X2 to have? Show that this
works in the case where n1 = n2 = 2 and p1 = p2 = 1/2. If n1 = n2 = 2, p1 = 1/2
and p2 = 1/8, show that X1 + X2 does not have a binomial distribution.
28. In a production line, it is estimated that 1 in every 200 items will be faulty. At the
quality control stage, each item is visually inspected, and it is believed that there is
a 90% chance of detecting a faulty item, and a 5% chance of mistakenly declaring a
non-faulty item as being faulty. In a batch of 10 items, what is the probability of two
items being declared faulty at the quality control stage? Define any notation that you
introduce, and justify your answer.
29. At a road junction, it is estimated that the mean number of car accidents is 5 per
year. Suggest a suitable probability distribution for the number of accidents at the
junction next year, and using your distribution, calculate the probability that there
will be more than 1 accident at the junction next year.
30. According to data collected by the British Geological Survey, earthquakes of magnitude
between 3 and 3.9 on the Richter scale occur in the UK on average (mean) 3 times
per year. Suppose we choose to model the number of such earthquakes occurring next
year with a Poisson distribution. Suggest a suitable rate parameter for the Poisson
distribution. Using this parameter value,
(a) what is the probability that there will be precisely 3 such earthquakes next year?
(b) What is the probability that there will be at least 2 such earthquakes next year?
31. Let X ∼ P oisson(5). If it is known that 4 ≤ X ≤ 6, what is the probability that
X = 5? Justify your answer.
32. On a roulette wheel, 18 numbers are coloured red, 18 numbers are coloured black, and
1 number is coloured green. A bet of £x on red will return £x plus the original stake
of £x (so that the profit is £x) if the ball lands on a red number.
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Consider the following betting strategy: bet £y on red, and if you win, stop betting. If
you lose, bet £2y on red. Then, if you win, stop betting, but if you lose, keep doubling
your stake until you win. From the table below, we see that if you win, your profit will
always be £y:
Win on attempt money spent (£)
1
y
y + 2y
2
3
y + 2y + 4y
y + 2y + 4y + 8y
4
..
..
.
.
Pn−1 a
n
a=0 2 y
(Recall that for a geometric series,
Px−1
j=0
payout (£)
2y
4y
8y
16y
..
.
profit
y
y
y
y
..
.
2 × 2n−1 y
y
krj = k(1 − rx )/(1 − r)).
(a) Calculate the expectation and standard deviation of the number of times that you
will bet on red.
(b) What is the probability that you will win within your first 20 attempts? In
practice, could you use this strategy to guarantee yourself a profit of £y?
33. Suppose you buy one National Lottery ticket each week, until you win a prize. Let Z
be the number of tickets you buy until you first win a prize, so that Z ∼ Geometric(p).
Clearly, losing one week doesn’t make it more likely that you will win the next week,
so which of the two statements below is true? Justify your answer with a proof.
(a) P (Z > z + n|Z > n) = P (Z > z + n),
(b) P (Z > z + n|Z > n) = P (Z > z).
(p is approximately 0.02, but you can leave your proof in terms of p).
34. * If Y ∼ P oisson(λ), prove that E(Y 2 ) = λ2 + λ. In your proof, you should not quote
the result that Var(Y ) = λ, but you may quote other results from the lecture notes.
35. * Let X ∼ P oisson(µ) and Y ∼ P oisson(µ), with X and Y independent. Derive the
probability mass function of Z = X + Y , and state the distribution of Z. You may
quote the result that
n
X
n!
,
(1 + 1)n =
a!(n
−
a)!
a=0
(which follows from the binomial theorem).
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36. Humans each carry 0, 1 or 2 copies of a particular gene. A woman who carries 1
copy of the gene has a child, whose paternity is disputed. Let X be the number of
copies of the gene carried by the child, and Y the number of copies carried by their
father. Standard genetic theory, plus background information about this particular
gene, suggest the following joint distribution
0
pX,Y (x, y)
0
16/50
x
1
16/50
2
0
y
1
2
4/50
0
8/50 1/50
4/50 1/50
(a) Calculate the marginal probability mass functions of X and Y .
(b) If it is known that the child carries only 1 copy of the gene, what would the
probability be that the father carries 2 copies? Justify your answer.
(c) Calculate the covariance between X and Y . Are X and Y independent? Justify
your answer.
37. Prove that Cov(X, Y ) = E(XY ) − E(X)E(Y ), and hence prove that Cov(X, Y + Z) =
Cov(X, Y ) + Cov(X, Z).
38. Let X and Y be the heights (in cm) of two adult sisters.
• Do you think X and Y should be independent?
• Do you think X − Y and X + Y should be independent?
If Var(X) = Var(Y ), find Cov(X + Y, X − Y ). Comment on your result.
39. Thirty students are enrolled on an MSc course. The possible grades that can be
awarded at the end of the course are fail, pass, merit and distinction. At the start
of the course, uncertainty about how many students will achieve each grade is to
be modelled with a multinomial distribution, and the expected numbers of students
achieving each grade are as follows.
fail pass merit distinction
2
8
15
5
For each question, define any notation that you introduce, and justify your answer.
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(a) Using the expected numbers of each grade, state suitable parameter values for
the multinomial distribution. How likely is it that the observed numbers of each
grade all equal to their expected values?
(b) What is the probability that more than one student fails?
(c) If four students fail, what is the probability that there will be 5 distinctions?
40. Three players play a game 4 times. Only one player can win each game, and there are
no draws. Let X denote the number of times player 1 wins, Y the number of times
player 2 wins and Z the number of times player 3 wins. Suppose that, in any single
game, player 1 wins with probability 0.5, player 2 wins with probability 0.2 and player
3 wins with probability 0.3.
(a) What is the joint probability distribution of X, Y, Z?
(b) Calculate the probability P (X = 2, Y = 1, Z = 1).
(c) Calculate the probability P (X > 2).
(d) Calculate the probability P (X > 2|Y = 1).
41. Humans each carry 0, 1 or 2 copies of a particular gene. Let X be the number of
copies of the gene carried by a child, and Y the number of copies carried by their
father. Standard genetic theory, plus background information about this particular
gene, suggest the following joint distribution
pX,Y (x, y)
0
0
16/50
x
1
16/50
2
0
y
1
2
4/50
0
8/50 1/50
4/50 1/50
In a sample of 10 randomly selected children, all with different fathers, what is the
probability that three children carry no copies, five children carry one copy, and two
children carry two copies of the gene? Justify your answer.
42. Let Z be a random variable with probability density function
3
3
fZ (z) = − z 2 + ,
2
2
for z ∈ [0, 1] and 0 otherwise.
10
(a) Tabulate the cumulative distribution function of Z.
(b) Give a check to show that your cumulative distribution function in part (a) is
correct.
(c) Calculate P (Z > 0.5).
(d) Find the expectation and standard deviation of Z.
43. A random variable X has cumulative distribution function given by

0
x ≤ 0,


 x2
0 ≤ x ≤ 1,
2
FX (x) =
x2

2x − 2 − 1 1 ≤ x ≤ 2,


1
x ≥ 2.
Explain how to obtain the probability density function of X, and then draw it. Include a check to show that your probability density function is correct. Calculate the
expectation and standard deviation of X.
44. Let Y be a random variable with probability density function
fY (y) = exp(−2|y|),
with RY = R.
(a) Calculate P (−1 < Y < 2).
(b) If it is known that Y > 0, calculate P (Y < 1|Y > 0). (Note that this is not the
same as P (0 < Y < 1)).
(c) Find the 75th percentile of the distribution of Y . (Hint: sketch the pdf of Y . If
y0.75 is the 75th percentile, what must the area under the pdf be between −∞
and y0.75 ?)
(d) Find the cumulative distribution function of Y .
45. Suppose X ∼ Exp(1). Without using a calculator, find the value of P (ln 1 ≤ X ≤ ln 2).
46. Radioactive decay can be modelled using an exponential distribution. It is estimated
that the half life of carbon-14 (used for radiocarbon dating) is 5730 years. This means
that the probability of one carbon-14 atom decaying into nitrogen-14 within 5730 years
is estimated to be 0.5. Find the expected time taken for one carbon-14 atom to decay
into nitrogen-14. Define any notation that you introduce, and justify your answer.
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47. A patient with gastroesophageal reflux disease is treated with a new drug to relieve
pain from heartburn. Following treatment, the time until the patient next experiences
the symptoms is recorded. The doctor treating the patient thinks there is a 50% chance
that the patient will stay symptom-free for at least 30 days. For each question, define
any notation that you introduce, and justify your answer:
(a) If the time until recurrence of symptoms is to be modelled using an exponential
distribution, find the rate parameter of this distribution based on the doctor’s
judgement.
(b) What is the probability the patient will remain symptom-free for at least 60 days?
(c) If, after 30 days, the patient has remained symptom-free, what is the probability
the patient will be symptom-free for at least another 30 days? (Note that the
answer to part (b) is different).
48. A cyclist leaves a bicycle chained to some railings, and returns five hours later to find
that the bike has been stolen. Define T to be the time in which the bike was stolen,
counting in hours from when the bike was left by the owner. Assuming that T has a
uniform distribution, calculate
(a) the mean of T and E(T 2 );
(b) the probability that T lies between 3 and 4 four hours;
(c) the 95th percentile of the distribution of T ;
(d) the probability that T = 2.
49. If W ∼ Exp(φ), prove that Var(W ) = 1/φ2 . You should not state that E(W 2 ) = 2/φ2
without proof. (Hint: you may use the result that if φ > 0, then x2 e−φx → 0 as
x → ∞).
50. (a) In R, generate one uniform U [0, 1] random variable using the command
runif(1,0,1). Denoting your observed value of the random variable by u, find
the u × 100th percentile of the Exp(0.1) distribution. Explain your method carefully. Check your calculation using the qexp command.
(b) In R, repeat the process in part (a) 100,000 times:
• generate 100,000 uniform U [0, 1] random variables, storing the result in a
vector u.
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• For each element in u, find the corresponding percentile of the Exp(0.1) distribution, using the command qexp(u,0.1), storing the results in a vector
y.
Plot a histogram of the 100,000 values in y using the command
hist(y,prob=T)
(the argument prob=T scales the histogram so that the total area of all the histogram bars is 1). Draw the density function of the Exp(0.1) distribution on top
of your histogram, using the command
curve(dexp(x,0.1),from=0,to=100,add=T)
Comment on your result.
51. (a) Let U ∼ U [0, 1] and let Y = U 2 . Calculate P (Y ≤ 0.09).
(b) Now suppose U ∼ U [0, 1] and let Y = − λ1 ln(1 − U ), where λ > 0. Derive an
expression for P (Y ≤ y). Comment on your result in relation to your results in
Q50.
52. Let X ∼ N (0.5, 0.25). Using the output from an R session below, calculate
(a) P (X > 1.5);
(b) P (0 < X < 1.5).
Note that two of the following four R commands are not relevant!
> pnorm(2,0,1)
[1] 0.9772499
> pnorm(1.5,0,1)
[1] 0.9331928
> pnorm(0,0,1)
[1] 0.5
> pnorm(1,0,1)
[1] 0.8413447
What commands would you use in R to calculate (a) and (b) directly?
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53. The total rainfall in Sheffield in January next year is to be modelled using a normal
distribution with mean 90mm and standard deviation 30mm. Using the result that
P (−1.645 < Z < 1.645) = 0.9,
where Z ∼ N (0, 1), calculate an interval (a1 , b1 ) such that total rainfall in Sheffield
in January has a 90% chance of lying in the interval. State a second interval (a2 , b2 )
that has (approximately) a 95% chance of containing the total rainfall in Sheffield in
January.
54. For this question, you will need the result that a linear combination of a set of independent normal random variables is another normal random variable (so Z defined below
has a normal distribution). Suppose X1 ∼ N (10, 18) and X2 ∼ N (10, 18), with X1 and
X2 independent. Calculate an interval (a1 , b1 ) such that P (Z ∈ (a1 , b1 )) ' 0.95, with
Z=
X1 + X 2
.
2
55. *
(a) Let Z ∼ N (0, 1). Find E(eZ ).
(b) Following your result in part (a), if ln Y ∼ N (µ, σ 2 ), is it true that E(Y ) = eµ ?
56. In a political opinion poll, 50 voters are asked whether they are in favour of a particular
policy, and 37 of them say that they are.
(a) What is the proportion of the people who are against the policy?
(b) If X is the random variable denoting the number of voters consulted in the opinion
poll that are in favour of the policy, give the distribution of X. Is this distribution
relevant only for this particular experiment?
57. The independent random variables X, Y, Z are normally distributed with E(X) =
E(Y ) = 1, E(Z) = 21 , Var(X) =Var(Y ) = 1 and Var(Z) = 1/8. Furthermore, two new
random variables U, W are defined as
U = 2X + 3Y − 8Z and
W = 2X − 2Y.
(a) Find the mean and variance of U and W .
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(b) Find the distribution of U and W .
58. 50 patients with gastroesophageal reflux disease are treated with a new drug to relieve
pain from heartburn. Following treatment, the time Ti until patient i next experiences
the symptoms is recorded. The doctor treating the patients thinks there is a 50%
chance that patient i will stay symptom-free for at least 30 days. Assuming the times
are independent and identically distributed, each with an exponential distribution, find
the expectation and variance of
50
T̄ (50) =
1 X
Ti .
50 i=1
59. A random variable has mean 2 and standard deviation 0.5. What can you say about
P ((X < −2) ∪ (X > 6))? What can you say about this probability if you learn that
the standard deviation is not 0.5, but is in fact 0.25?
60. * Prove Markov’s inequality, that for a continuous random variable X with range [0, ∞)
P (X > c) ≤
E(X)
,
c
for any c > 0.
61. Use the moment generating function for X ∼ N (µ, σ 2 ) to check that E(X) = µ, and
Var(X) = σ 2 . Find E(X 3 ) and E(X 4 ) using this technique.
62. Suppose that the random variable X has an exponential distribution with rate λ. Write
down the moment generating function for the random variable −X. Compute E(X n ),
and check earlier calculations of the mean and variance of X.
The reason for working with −X rather than X, is that if t ≥ 0 (and this is in fact
sufficient) then all the integrals are finite.
63. Let X ∼ Bin(n, p). Starting from the probability function, find the moment generating
function of X and use this technique to confirm that X is the sum of n i.i.d. random
variables each of which has a Bernoulli(p) distribution.
64. (a) A continuous random variable X having pdf f , has moment generating function
MX . If a and b are arbitrary real numbers, show that the moment generating
function of the random variable aX + b is given for each t ∈ R by
MaX+b (t) = etb MX (at).
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(b) If X is a random variable (discrete or continuous), what can you say about MX (0)?
65. Suppose that each week, the number of calls to a minicab firm has a Poisson distribution
with rate parameter 100. Use the central limit theorem to estimate the probability that
the total number of calls in one year is no more than 5000.
66. (a) Suppose that the random variable X ∼ P oisson(λ). Show that
MX (t) = exp {λ(et − 1)}.
(b) If Sn is the sum of n i.i.d. random variables, each of which is P oisson(λ), show
that S(n) ∼ P oisson(nλ).
(c) Using the result of (b), revisit question 65 and find a suitable R command to
calculate the probability directly, without using the CLT.
67. Using a particular instrument, a measurement X of the mass (in grams) is expressed
as X = µ + 2(U − 12 ), where U follows the uniform distribution U (0, 1) and µ is its
unknown true weight. Calculate the mean and variance of X. You can assume that µ
will be much greater than zero, so you need not worry about measurements being less
than zero.
(a) Five measurements X1 , . . . , X5 are taken of the mass of the same object, each
with the same distribution as X above. Find the approximate distribution of
X̄ = 51 (X1 + X2 + X3 + X4 + X5 ). Do you expect this approximation to work
well? Justify your answer.
(b) Carry out some simulation experiments in R to investigate the histogram of X̄ in
(a). Experiment with different sample sizes n (instead of 5, try 10, 100, etc). For
what sample size does the distribution of the X̄ start to look different from the
distribution of Xi themselves? What size sample is ‘large’, in the sense of practical
use of the central limit theorem? You will need to use the function runif in R,
and fix some arbitrary value for µ.
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Hints to selected questions
3. Try writing D = B ∪ C, then consider how to expand m(A ∪ D) using the results in
your notes and in the question.
4. For a function g to be a measure, what requirements must it satisfy? We are told that
m and n are measures, so for any A, B ∈ S with A ∩ B = ∅, what do we know about
m(A ∪ B) and n(A ∪ B)?
5. If P is a probability measure, what is the value of P (S)? What is the relationship
between S, A and Ā?
16. Good notation is important; using poor notation makes this problem much harder to
solve! Examples of poor notation are as follows.
1) The question is asking us for
P (5|1) =
P (5 ∩ 1)
.
P (1)
This notation is very confusing. What is 5 ∩ 1? If it’s the empty set, aren’t we going
to end up with P (5|1) = 0? Is P (1) the (prior) probability that there is exactly one
forgery (0.09), or the probability of randomly selecting one painting and discovering
that it is a forgery?
2) Let A be the event of 5 forgeries, and B be the event that the selected
painting is forgery. We want
P (A|B) =
P (A)P (B|A)
.
P (B)
This is better, but notation masks the fact that there could be other numbers of
forgeries, which is important for calculating P (B).
An example of better notation is
Let Ei be the event that there are exactly i forgeries, and F be the event that
the randomly selected painting is a forgery.
The question is asking for P (E5 |F ). From the definitions of Ei and F , what is P (F |Ei )?
Now consider how to use Bayes’ theorem to get P (E5 |F ).
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17. As with Q16, good notation helps. Let EA , EB and EC be the events that prisoners A,
B and C are to be executed, respectively. Suppose the jailer names prisoner B as being
set free. Let F be the event that the jailer chooses to name B as being set free. This
event is not the same as EB . We know P (EA ) = P (EB ) = P (EC ) = 1/3. Consider the
values of P (F |EA ), P (F |EB ) and P (F |EC ), then try using Bayes’ theorem.
24. First suppose that p = 0.6. Do you maximise your expected earnings by choosing
q = 0.6? Now suppose p = 0.4. Do you maximise your expected earnings by choosing
q = 0.4? Now consider how you should choose q, dependent on whether p > 0.5.
27. When n1 = n2 = 2, X1 + X2 can take integer values from 0 to 4. Consider the different
ways it can take each of these values in terms of the values of X1 and X2 , and calculate
their probabilities.
34. The idea is similar to that used to obtain the mean of a Poisson random variable:
change variables in the sum to make it look like something we know the answer to.
35. Consider, for example, pZ (1) = P (Z = 1). If Z = 1, what values could X and Y
take, such that Z = X + Y = 1? The only possibilities are (X = 0, Y = 1) and
(X = 1, Y = 0), so that
pZ (1) = P (X = 0, Y = 1) + P (X = 1, Y = 0),
As X and Y are independent, we have
pZ (1) = P (X = 0)P (Y = 1) + P (X = 1)P (Y = 0) = pX (0)pY (1) + pX (1)pY (0).
Now consider the general case, pZ (z). If Z = z, what possible values could X and Y
take? Try writing an
P expression for pZ (z) in terms of the probability mass functions
of X and Y , using
notation, and then see how you can simplify your expression.
55. Start with the definition of E(g(X)), tidy up the integrand as far as you can, then
consider the following:
(a) We can write x2 − 2x as (x − 1)2 − 1.
(b) What is the density function of a N (1, 1) random variable? What do we get when
we integrate this function between −∞ and ∞?
60. Study the proof of Chebyshev’s inequality, and try something similar using the mean
of X rather than the variance.
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