Section 4 (MATH265)
Probability. .
Abbreviation: RV means Random Variable.
wrt means with respect to.
Integration with respect to probability distributions
Theorem. Given a RV X on a probability space (, F, P ) and a Borel
function g : R R,
Z
Z
g(X()dP ()
Section 1 (MATH265)
Set Theory.
Notations:
N = cfw_1, 2, . . . collection (or the set) of all natural numbers;
Z = cfw_. . . , 2, 1, 0, 1, 2, . . . collection of all integers;
n
Q collection of all rationals, i.e. numbers of the form m
, where n, m
Z, m
Section 3 (MATH265)
Measurable Functions. .
Abbreviation: RV means Random Variable.
The domain of the functions we shall be considering is usually R (or a
measurable subset E R), as well as the range.
Definition. We say that a function f :
measurable if f
Section 5 (MATH265)
Product measures. .
Definition. A (standard) rectangle in R2 is the cartesian product R =
I1 I2 , where I1 , I2 are intervals. The area (or the length) of a rectangle is
a(R) = l(I1 ) l(I2 ).
If we take the measurable spaces (1 = R, F1
MATH265 Measure Theory
Feedback on Exercise Sheet 3.
Problem 1. We need to say that, for some positive , the statement
N N : n > N |fn A| <
(1)
is false. Therefore, the required formula begins with > 0 . . . . Statement (1) is false if, for all
N N, the
MATH265 Measure Theory
Feedback on Exercise Sheet 5.
Problem 1. For a fixed irrational x [0, 1], we need to prove that
> 0 > 0 : y [0, 1] |x y| < = |f (y)| < .
1
< . For any fixed
(Note that x > 0.) For an arbitrarily fixed > 0, take M such that M
m
m =
MATH265 Measure Theory
Feedback on Exercise Sheet 10.
Problem 1. (a) Solve the given equations X =
1 =
(X + Y )2
;
4
1 + 2 ; Y =
2 = (X Y )/2.
1 2 for 1 , 2 :
()
The range of (X, Y ) is defined by inequalities 1 [0, 1], 2 [0, 1]:
0
(x + y)2
1 = 0 x + y 2
Section 2 (MATH265)
Measure. .
Null sets
Suppose I is a bounded interval of any kind, i.e. I = [a, b], I = [a, b),
I = (a, b], or I = (a, b) with a b. We define the length of I (the measure
of I) as l(I) = b a in each case. In particular l(cfw_a) = l([a,
MATH265 Measure Theory
Feedback on Exercise Sheet 8.
Problem 1. (a) For any B B, PX (B) = 0 if B does not contain any natural numbers (or
zero), and PX (B) is positive otherwise. This means
PX (B) = p0 0 (B) + p1 1 (B) + p2 2 (B) + . . . ,
where is the Di
MATH265 Measure Theory
Feedback on Exercise Sheet 4.
Problem 1. Yes, F is a -field. We need to check the axioms:
(i) The whole set R is a full set; hence R F.
(ii) If B F then either B N and hence B c L B c F, or B L and hence
c
B N B c F,
[
(iii) Suppose
MATH265 Measure Theory
Feedback on Exercise Sheet 9.
Problem 1. L1 (E) is the space of functions integrable over E. The (indefinite) integral, or
antiderivative, has the form:
1 +1
Z
, if 6= 1;
+1 x
x dx =
ln x,
if = 1.
1
(a) One can substitute the limi
MATH265 Measure Theory
Feedback on Exercise Sheet 6.
Problem 1. (a) Fix an arbitrary Borel set B B and introduce the set
4
C = cfw_y : f (y) B = f 1 (B).
Obviously,
h1 (B) = cfw_x : g(x) C = g 1 (C).
The set C is Borel-measurable because f is Borel-measur
MATH265 Measure Theory
Feedback on Exercise Sheet 7.
Problem
1. Since the set Q of rational numbers is denumerable, f = 0 almost everywhere.
Z
f dm = 0.
Hence
E
Problem 2. (a) For odd n we have fn (x) = I[0, 1 ] (x); for even n fn (x) = I[ 1 ,1] (x). In a
MATH265 Measure Theory
Feedback on Exercise Sheet 2.
Problem 1. We need to construct a 1-1 correspondence between the union
A=
[
An = cfw_a11 , a21 , . . . , an1 1 , a12 , a22 , . . . , an2 2 , a13 , a23 , . . . , an3 3 , a14 , . . .
i=1
and the set N = c
MATH265 Measure Theory
Feedback on Exercise Sheet 1.
Problem 1. Let m N. Then m does not belong to the subset Am+1 = cfw_m + 1, m + 2, . . .
and does not belong to A. Hence A contains no elements: A = (empty set).
Problem 2. If a > 0 then, for some n N, a
m
Math362 Applied Probability: Assignment 1
Hand-in: not later than 9:50pm, Wednesday, 14th October.
1.1. In a family of two children, each independently a boy with probability 0.5, dene the
1.2.
1.3.
sample space and relevant events and calculate the pro
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PAPER CODE NO.
MATH 362
EXAMINER:
Gashi/Papaioannou, TEL.NO. 44024/50141
DEPARTMENT: Mathematical Sciences
JANUARY 2015 EXAMINATIONS
APPLIED PROBABILITY
Time allowed: Two and a half hours
INSTRUCTIONS TO CANDIDATES:
All seven questions carry equal weight.
Math362 Applied Probability: Assignment Solution 6
6.1. a) Let T, be the rst time the Markov Chain visits state i i.e T, = mincfw_n > 0 : X, = 3'. Set
f3=P(Z-; <00|X0=i),
m.=E[n|X0=r].
State 3' is recurrent if f, = 1. (This means that the state Will be vi
Math362 Applied Probability: Assignment 6
Hand-in: not later than 9:50am, Wednesday, 25th November.
6.1. a) State the definitions of positive recurrent, null recurrent and transient states and
discuss intuitive meanings of these definitions.
b) Suppose cf
Math362 Applied Probability: Assignment 7
Hand-in: not later than 9:50am, Wednesday, 2nd December.
7.1. The no claims discount scale operated by a motor insurer has three levels of discount
0%, 30% and 50%. The rules for moving between discount levels are
Math362 Applied Probability: Assignment Solution 2
2.1. (a) Let us calculate the conditional probability mass function of X given that X + Y = n.
We obtain
P (X = k, X + Y = n)
P (X = k | X + Y = n) =
P (X + Y = n)
P (X = k, Y = n k)
=
P (X + Y = n)
P (X
Math362 Applied Probability: Assignment Solution 3
3.1. (i) Let X be the number of suits that Rebecca sells, and N be the number of customers who
enter the store. Then
P cfw_X = 0 =
=
=
=
X
P cfw_X = 0 | N = nP (N = n)
n=0
X
P cfw_X = 0 | N = n
n=0
X
(1 p
Math362 Applied Probability: Assignment 1
Hand-in: not later than 9:50pm, Wednesday, 14th October.
1.1. In a family of two children, each independently a boy with probability 0.5, define the
sample space and relevant events and calculate the probability t
Math362 Applied Probability: Assignment Solution 4
4.1. a) We will compute, for example, P (X2 = 3):
P (X2 = 3) =
5
X
P (X2 = 3 | X1 = i)P (X1 = i);
i=1
P (X1 = i) =
5
X
P (X1 = i | X0 = j) =
j=1
o
1n
P (X1 = i | X0 = 1) + P (X1 = i | X0 = 5)
2
(P (X0 = 2
Math362 Applied Probability: Assignment 10
Hand-in: not later than 9:50am, Wednesday, 23th December.
10.1. A machine suffers instantaneous stoppages of two kinds, major and minor, according
to independent Poisson processes with rates and respectively. Let
Math362 Applied Probability: Assignment 5
Hand-in: not later than 9:50am, Wednesday, 18th November.
5.1. Classify the states of the following DTMCs (i.e. identify the communicating classes,
state whether they are periodic or aperiodic, recurrent or transi
Math362 Applied Probability: Assignment Solution 10
10.1.
P (N = n | T = t) = P (n minor stoppages in [0, t)
since processes are independent
= et
(t)n
for n = 0, 1, 2, , t 0.
n!
Therefore,
Z
P (N = n | T = t)fT (t)dt
P (N = n) =
0
(continuous version of l