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Title: Probability and Statistics
Description: Subject content are for 3rd year students. random experiment, sample space, events and probability of event.
Description: Subject content are for 3rd year students. random experiment, sample space, events and probability of event.
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Chapter 1
Introduction
Lectures 1 - 3
...
By a random experiment, we mean an experiment which has multiple outcomes and one don't know in
advance which outcome is going to occur
...
We assume
that the set of all possible outcomes of the experiment is known
...
1
...
Example 1
...
1 Toss a coin and note down the face
...
More over, the sample space is
Example 1
...
2 Toss two coins and note down the number of heads obtained
...
Example 1
...
3 Pick a point `at random' from the interval
picking any point
...
`At random' means there is no bias in
...
2 ( Event ) Any subset of a sample space is said to be an event
...
0
...
0
...
Definition 1
...
If
and
are mutually exclusive, then occurrence of
Note that non occurrence of
Example 1
...
5 The events
But the events
,
implies non occurrence of
need not imply occurrence of
,
, since
and vice versa
...
0
...
are not mutually exclusive
...
Intuitively
probability quantifies the chance of the occurrence of an event
...
In general it is not possible to assign probabilities to all events from the
sample space
...
0
...
So one need to restrict to a smaller class of subsets of the sample space
...
0
...
Therefore, one can assign probability to any finite union of intervals in
, by
representating the finite union of intervals as a finite disjoint union of intervals
...
Also note that if one can assign probability to an
event, then one can assign probability to its compliment, since occurence of the event is same as the
non-occurance of its compliemt
...
This leads to the following
special family of events where one can assign probabilities
...
4 A family of subsets
following
...
0
...
e
...
Moreover, if
is the smallest and
Then
Let
is a
Lemma 1
...
1
is a
is the largest
be a nonempty set and
-field and is the smallest
is called the
is a
be a nonempty set
...
0
...
-field generated by
Let
-field of subsets of
-field of subsets of
, then
...
Define
-field containing the set
...
be an index set and
be a family of
-fields
...
Proof
...
Now,
Similarly it follows that
Hence
is a
-field
...
0
...
Then
-field and is the smallest
-field containing
This can be seen as follows
...
0
...
If
-field containing
(ii) if
, then
-field containing
is a
, then
-field
...
Hence,
...
5 A family
(i)
is a
...
0
...
-field is a field
...
0
...
...
Note that (i) and (ii) in the definition of field follows easily
...
i
...
,
To see that
and if either
or
, if both
is finite, then
is finite
...
is not a
-field, take
Now
Definition 1
...
A map
is said to be
a probability measure if P satisfies
(i)
(ii) if
are pairwise disjoint, then
Definition 1
...
; where
The triplet
, a nonempty set (sample space),
,a
-field and
, a probability
measure; is called a probability space
...
Define
on
as follows
...
0
...
This probability space corresponds to the random experiment of
tossing an unbiased coin and noting the face
...
0
...
Solution
...
Define
on
as follows
...
Then
's are disjoint)
Therefore
Therefore properties (i) , (ii) are satisfied
...
0
...
Then
...
Since
,
is a probability measure
...
Now
Therefore
since
...
We prove (5) by induction
...
Hence by induction property (5) follows
...
Now we prove (6)(i)
...
0
...
0
...
0
...
0
...
0
...
Note that
Now using (6)(i) we have
i
...
,
Hence
From property (2), it follows that
i
...
,
Therefore
(1
...
4)
Set
Then
and are in
...
0
...
Using (1
...
5), letting
(1
...
4), we have
in
Recall that all the examples of probability spaces we had seen till now are with sample space finite or
countable and the
-field as the power set of the sample space
...
Consider the random experiment in Example 1
...
3, i
...
Since point is picked 'at random', the probability measure should satisfy the following
...
0
...
...
0
...
Set
...
then
can be represented as
where
,
Then
where
Therefore
...
is a field
...
(1
...
7)
where
's are pair wise disjoint intervals of the form
Extension of
from
to
...
To understand the
statement of the extension theorem, we need the following definition
...
8 (Probability measure on a field) Let
be a nonempty set and
be a field
...
0
...
0
...
0
...
Using Theorem 1
...
2, one can extend
defined by (1
...
7) to
see Exercise 1
...
Since
on
preserving (1
...
6)
...
Title: Probability and Statistics
Description: Subject content are for 3rd year students. random experiment, sample space, events and probability of event.
Description: Subject content are for 3rd year students. random experiment, sample space, events and probability of event.