M3S3/M4S3 : SOLUTIONS 2
1. To establish a.s. convergence, apart from considering the original denition directly, we might
consider three possible methods of proof;
I the equivalent characterization
a.s.
Xn X
lim P [ |Xm X| < , m n ] = 1
for each
n
> 0.
II

1
M3/M4S3 STATISTICAL THEORY II
LIMITS FOR REAL FUNCTIONS
Denition : Limits
Let f be a real-valued function of real argument x.
Limit as x :
f (x) a
as
x
or
lim f (x) = a
x
if, for all > 0, M = M () such that |f (x) a| < , x > M
Limit as x x :
0
f (x)

1
M3/M4S3 STATISTICAL THEORY II
MEASURABLE FUNCTIONS
The real-valued function f dened with domain E , for measurable space (, F), is Borel
measurable with respect to F if the inverse image of set B, dened as
f 1 (B) cfw_ E : f () B
is an element of -algeb

M3S3/M4S3
ASSESSED COURSEWORK 2 : SOLUTIONS
1. The likelihood for is
n
Ln () =
n
fX| (xi |) =
i=1
i=1
1
xi
exp
xi
n
= n t1 exp
n
1
n
x
i
i=1
n
where tn =
xi , and hence the log-likelihood is
i=1
1
ln () = n log n log + ( 1) log tn
n
x .
i
i=1
For t

M3S3/M4S3
ASSESSED COURSEWORK 1 : SOLUTIONS
1. We use the result from the handout
n
X(k1 )
X(k2 )
xp1
xp2
p1 (1 p1 )
cfw_fX (xp1 )2
N 0,
L
p1 (1 p2 )
fX (xp1 ) fX (xp2 )
p1 (1 p2 )
fX (xp1 ) fX (xp2 )
p2 (1 p2 )
cfw_fX (xp2 )2
which implies,

IMPERIAL COLLEGE LONDON
UNIVERSITY OF LONDON
BSc and MSci EXAMINATIONS (MATHEMATICS)
MAYJUNE 2005
M3S3/M4S3 (SOLUTIONS)
STATISTICAL THEORY II
c 2005 University of London
M3S3/M4S3 (SOLUTIONS)
Page 1 of 10
1. (a) (i) Let be a set, and F be a set of subsets

M3S3/M4S3 : SOLUTIONS 3
1. (a) Using the hint given; we know, by properties of vector random variables,
k
ai Xi = V ar aT X = aT a
V ar[Y ] = V ar
i=1
where variances taken with respect to the distribution of Y and X on the left and right hand sides
respe

SAMPLE EXAM QUESTIONS - SOLUTION
As you might have gathered if you attempted these problems, they are quite long relative to the 24
minutes you have available to attempt similar questions in the exam; I am aware of this. However, these
questions were desi

M3S3/M4S3 : SOLUTIONS 4
1. Equate terms on the two sides of the equation yields the constants, that is, compare coecients of
terms in x2 , x and 1 as follows:
x2 : A + B = C
x : 2Aa + 2Bb = 2Cc
2
2
c=
2
1 : Aa + Bb = Cc + d
and as
Aa + Bb
Aa + Bb
=
C
A+B

M3S3/M4S3 : SOLUTIONS 1
1. First let E1 E2 E3 . . . is an increasing sequence of sets. Using the decomposition given,
En+1 En En+1 En
it follows that
lim En
n
En E1
n=1
Fn
n=2
where
Fn En En1 .
But, the cfw_Fn are a sequence of mutually exclusive event

1
M3/M4S3 STATISTICAL THEORY II
INTEGRAL WITH RESPECT TO MEASURE
Let (, F, ) be a measure space, and be a non-negative simple function, : R , that
is, for ,
k
() =
ai IAi ()
i=1
for real constants a1 , ., ak 0 and measurable sets A1 , ., Ak F, for some k