1. (a) Explain what is meant by a eld A of subsets of a sample space .
Let A and B belong to some eld A. Show that A contains the set A B.
What is meant by a probability space?
(b)
Let X and Y have joint probability density of the form
fX,Y (x, y) =
cx2 y
1
M1S : EXERCISE SHEET 3
CONDITIONAL PROBABILITY AND THE PROBABILITY THEOREMS
1. Given an event F in sample space with P (F ) > 0, the conditional probability operator, dened on the
events of , satises the probability axioms (I), (II) and (III), that is,
1
M1S : EXERCISE SHEET 2 : SOLUTIONS
1.(a) Let C = event person is correctly classied. Partition according to disease status, event person is misclassied
is C = (S T ) (S T )
(b) Partition rst into correctly classied suerers/non-suerers
(i)
(i)
(D (X T )
1
M1S : EXERCISE SHEET 5 : SOLUTIONS
1. (i) X Binomial(500, 1/365)
fX (x) =
500
x
x
1
365
1
1
365
500x
x = 0, 1, 2, .500
(ii) If = 500/365, then
x
=
500!
1
x!(500 x)! 365
=
1
500!
1/365
x! (500 x)! 1 1/365
1
1
500x
x!
365
fX (x)
500x
1
365
x e
x!
1
x
1
x
1
M1S : EXERCISE SHEET 1 : SOLUTIONS
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
AB C
ABC
ABC
ABC
(A B C ) (A B C) (A B C) (A B C)
(A B C ) (A B C ) (A B C)
(A B C ) (A B C) (A B C)
A B C
(A B C) A B C
2. (a)
(b)
(c)
(d)
(e)
TRUE
TRUE
TRUE
FALSE
TRUE
3. (a)
(b
1
M1S : EXERCISES 10
JOINT DISTRIBUTIONS AND SUMS OF RANDOM VARIABLES
1. Two fair dice are rolled. Let X and Y be the discrete random variables corresponding to the largest and
smallest of the two scores respectively. The pair (X, Y ) is a vector of rando
1
M1S : EXERCISE SHEET 4 : SOLUTIONS
1. (i) Special case of the Binomial Expansion of (a + b)n with a = 1, b = 1.
(ii) (x + 1)m+n = (x + 1)m (x + 1)n , and
m+n
L.H.S.
: (x + 1)m+n
m+n
k
=
k=0
m
R.H.S.
: (x + 1)m (x + 1)n
=
i=0
m
m
i
n
i=0 j=0
k
=
k=0
i=0
1
M1S : EXERCISES 4
COMBINATORICS : HYPERGEOMETRIC PROBABILITIES AND PARTITIONING
1. The binomial expansion is used to express (a + b)n in power series form:
n
(a + b)n =
k=0
n
k
ak bnk .
Use the binomial expansion to prove the following identities:
For n
1
M1S : EXERCISE SHEET 6 : SOLUTIONS
1. Need to show fX non-negative, and sums to one over the range of X. Sum of geometric progression gives
result, that is,
x
1
1
1
1
=
=1
fX (x) =
1+ 1+
1+
1+
x=0
x=0
Distribution function FX given, for x = 0, 1, . by
x
1
M1S : EXERCISE SHEET 7 : SOLUTIONS
1. (i) Density function must integrate to 1 over X = [0, 1], so
1
1
cx2 + x dx = 1 = c
fX (x) dx = 1 =
0
0
x3
x2
+
3
2
1
= 1 = c =
0
3
2
(ii) Distribution function FX given for 0 x 1 by
x
FX (x) =
fX (t) dt =
0
x3 + x2
1
M1S : EXERCISE SHEET 8 : SOLUTIONS
1. (i) By an elementary probability result
RX (x) = P [X > x] = 1 P [X x] = 1 FX (x)
(ii) By denition, and from (i),
hX (x) =
fX (x)
fX (x)
=
RX (x)
1 FX (x)
(1)
and integrating both sides with respect to x gives
HX (x
1
M1S : EXERCISES 8
CONTINUOUS PROBABILITY DISTRIBUTIONS : PROPERTIES AND FUNCTIONALS
1. Probability models for random variables taking values on R+ may be specied via a number of probability
functions: for real valued x
Continuous CDF
FX (x) = P [X x]
PD
1
M1S : EXERCISES 9
TRANSFORMATIONS OF RANDOM VARIABLES
1. Suppose that X is a continuous random variable with range X = [0, 1], and probability density function fX
specied by
fX (x) = 2(1 x)
0x1
and zero otherwise. Find the probability distributions of r
1
M1S : EXERCISE SHEET 2
ELEMENTARY PROBABILITY
1. A person selected from a population and subjected to screening-for-a particular-disease is either a suerer (S)
or not (S ). The screening will either result in a positive test (T ), or a negative (T ). If
2.3. CONTINUOUS RANDOM VARIABLES
17
and zero otherwise. For x 1, let k(x) be the largest integer not greater than x, then
k(x)
FX (x) =
fX (xi ) =
xi x
i=1
fX (i) = 1
k(x)
1
2
and FX (x) = 0 for x < 1.
Graphs of the probability mass function (left) and c
1.8. COUNTING TECHNIQUES
9
NOTE : The items may be distinct (unique in the collection), or indistinct (of a unique type in
the collection, but not unique individually).
e.g. The numbered balls in the National Lottery, or individual playing cards, are dist
CHAPTER 1
DEFINITIONS, TERMINOLOGY, NOTATION
1.1
EVENTS AND THE SAMPLE SPACE
Denition 1.1.1 An experiment is a one-o or repeatable process or procedure for which
(a) there is a well-dened set of possible outcomes
(b) the actual outcome is not known with c
6.3. MODES OF STOCHASTIC CONVERGENCE
6.3
6.3.1
57
MODES OF STOCHASTIC CONVERGENCE
CONVERGENCE IN DISTRIBUTION
Denition 6.3.1 Consider a sequence cfw_Xn , n = 1, 2, . . ., of random variables and a
corresponding sequence of cdfs, FX1 , FX2 , . so that for
2.8. JOINT PROBABILITY DISTRIBUTIONS
33
Example 2.6 We record the delay that a motorist encounters at a one-way trac stop sign. Let
X be the random variable representing the delay the motorist experiences. There is a certain
probability that there will be
7.3. HYPOTHESIS TESTING
65
In the Normal example given above, we have that:
z is the test statistic;
The distribution of random variable Z if H0 is true is the null distribution;
= 0.05 is the signicance level of the test (choosing = 0.01 gives a stron
2.6. TRANSFORMATIONS OF RANDOM VARIABLES
25
and, by dierentiation, because g is monotonic increasing,
fY (y) = fX (g 1 (y)
d
g 1 (t)
dt
t=y
= fX (g 1 (y)
d
g 1 (y)
dt
t=y
,
as
d
g 1 (t) > 0.
dt
Case (II): If g is decreasing, then for x X and y Y we have
g
CHAPTER 3
DISCRETE PROBABILITY DISTRIBUTIONS
Denition 3.1.1 DISCRETE UNIFORM DISTRIBUTION
X U nif orm(n)
fX (x) =
1
,
n
x X = cfw_1, 2, ., n ,
and zero otherwise.
Denition 3.1.2 BERNOULLI DISTRIBUTION
X Bernoulli()
fX (x) = x (1 )1x ,
x X = cfw_0, 1 ,
and
49
Taking limiting cases of the Student-t distribution
n : St(n) N (0, 1),
n 1 : St(n) Cauchy.
7. If X1 N (1 , 2 ) and X2 N (2 , 2 ) are independent and a, b are constants, then
1
2
T = aX1 + bX2 N (a1 + b2 , a2 2 + b2 2 ).
1
2
50
CHAPTER 4. CONTINUOUS PR
1
M1S : EXERCISE SHEET 1
EVENTS AND SAMPLE SPACES
1. Let A, B and C be three arbitrary events. Using only the operations of union, intersection and complement,
write down expressions for the following events:
(a) Only A occurs.
(c) All three events occur.
1
M1S : EXERCISES 5
DISCRETE PROBABILITY DISTRIBUTIONS
1. (i) The birthdays of 500 pupils in a school are recorded. Let X be the discrete random variable corresponding
to the number of these pupils who have their birthday on New Years Day.
Assuming that b
1
M1S : EXERCISES 7
CONTINUOUS PROBABILITY DISTRIBUTIONS
1. The length of time (in hours) that a student takes to complete a one hour exam is a continuous random
variable X with probability density function fX dened by
fX (x) = cx2 + x
0<x1
for some const
1
M1S : EXERCISES 6
PROPERTIES OF DISCRETE PROBABILITY DISTRIBUTIONS
1. Show that the function
1
1+
x
1+
for parameter > 0 is a valid probability mass function for a discrete random variable X taking values on
cfw_0, 1, 2, ., and nd the corresponding cumu