Revision notes  ST2131
Ma Hongqiang
April 18, 2017
Contents
1
Combinatorial Analysis
2
2
Axioms of Probability
3
3
Conditional Probability and Independence
5
4
Random Variables
7
5
Continuous Random Variable
12
6
Jointly Distributed Random Variables
18
7
Properties of Expectation
24
8
Limit Theorems
29
9
Problems
30
1
1
Combinatorial Analysis
Theorem 1.1
(Generalised Basic Principle of Counting)
.
Suppose that
r
experiments are to be preformed. If
•
experiment 1 can result in
n
1
possible outcomes;
•
experiment 2 can result in
n
2
possible outcomes;
• · · ·
•
experiment
r
can result in
n
r
possible outcomes;
then together there are
n
1
n
2
· · ·
n
r
possible outcomes of the
r
experiments.
1.1
Permutations
Theorem 1.2
(Permutation of distinct objects)
.
Suppose there are
n
distinct objects, then the total number of permutations is
n
!
.
Theorem 1.3
(General principle of permutation)
.
For
n
objects of which
n
1
are alike,
n
2
are alike,
. . .
,
n
r
are alike, there are
n
!
n
1
!
n
2
!
· · ·
n
r
!
different permutations of the
n
objects.
1.2
Combinations
Theorem 1.4
(General principle of combination)
.
If there are
n
distinct objects, of which we choose a group of
r
items, number of combinations
equals
n
r
=
n
!
r
!(
n

r
)!
1.2.1
Useful Combinatorial Identities
1. For 1
≤
r
≤
n
,
(
n
r
)
=
(
n

1
r

1
)
+
(
n

1
r
)
2. (
Binomial Theorem
) Let
n
be a nonnegative integer, then
(
x
+
y
)
n
=
n
X
k
=0
n
k
x
k
y
n

k
3.
∑
n
k
=0
(
n
k
)
= 2
n
4.
∑
n
k
=0
(

1)
k
(
n
k
)
= 0
2
1.3
Multinomial Coefficients
If
n
1
+
n
2
+
· · ·
+
n
r
=
n
, we define
(
n
n
1
,n
2
,
···
,n
r
)
by
n
n
1
, n
2
,
· · ·
, n
r
=
n
!
n
1
!
n
2
!
· · ·
n
r
!
Thus
(
n
n
1
,n
2
,
···
,n
r
)
represents the number of possible divisions of
n
distinct objects into
r
distinct groups of respective sizes
n
1
, n
2
,
· · ·
, n
r
.
Theorem 1.5
(Multinomial Theorem)
.
(
x
1
+
x
2
+
· · ·
+
x
r
)
n
=
X
(
n
1
,...,n
r
):
n
1
+
···
+
n
r
=
n
n
n
1
, n
2
,
· · ·
, n
r
x
n
1
1
x
n
2
2
· · ·
x
n
r
r
1.4
Number of Integer Solutions of Equations
Theorem 1.6.
There are
(
n

1
r

1
)
distinct
positive
integervalued vectors (
x
1
, x
2
, . . . , x
r
) that
satisfies the equation
x
1
+
x
2
+
· · ·
+
x
r
=
n
where
x
i
>
0 for
i
= 1
, . . . , r
Theorem 1.7.
There are
(
n
+
r

1
r

1
)
distinct
nonnegative
integervalued vectors (
x
1
, x
2
, . . . , x
r
)
that satisfies the equation
x
1
+
x
2
+
· · ·
+
x
r
=
n
where
x
i
>
0 for
i
= 1
, . . . , r
2
Axioms of Probability
2.1
Sample Space and Events
The basic objects of probability is an
experiment
: an activity or procedure that produces
distinct, welldefined possibilities called
outcomes
.
The
sample space
is the set of all possible outcomes of an experiment, usually denoted by
S
.
Any subset
E
of the sample space is an
event
.
2.2
Axions of probability
Probability
, denoted by
P
, is a function on the collection of events satisfying
(i) For any event
A
,
0
≤
P
(
A
)
≤
1
(ii) Let
S
be the sample space, then
P
(
S
) = 1
3
(iii) For any sequence of mutually exclusive events
A
1
, A
2
, . . .
(that is
A
i
A
j
=
∅
when
i
6
=
j
)
P
(
∪
∞
i
=1
A
i
) =
∞
X
i
=1
P
(
A
i
)
2.3
Properties of Probability
Theorem 2.1.
P
(
∅
) = 0.
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