Maths Probability lec1/8.pdf

7/27/2019 Maths Probability lec1/8.pdf

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Chapter 1

Introduction

Lectures 1 - 3.

n this chapter we introduce the basic notions random experiment, sample space, events and probabilityof event.By a random experiment, we mean an experiment which has multiple outcomes and one don't know inadvance which outcome is going to occur. We call this an experiment with `random' outcome. We assumehat the set of all possible outcomes of the experiment is known.

Definition 1.1. Sample space of a random experiment is the set of all possible outcomes of the randomexperiment.

Example 1.0.1 Toss a coin and note down the face. This is a random experiment, since there aremultiple outcomes and outcome is not known before the toss, in other words, outcome occur randomly.

More over, the sample space is

Example 1.0.2 Toss two coins and note down the number of heads obtained. Here sample space is

.

Example 1.0.3 Pick a point `at random' from the interval . `At random' means there is no bias in

picking any point. Sample space is .

Definition 1.2 ( Event) Any subset of a sample space is said to be an event.

Example 1.0.4 is an event corresponding to the sample space in Example 1.0.1.

Definition 1.3 (mutually exclusive events) Two events are said to be mutually exclusive if

.

f and are mutually exclusive, then occurrence of implies non occurrence of and vice versa.

Note that non occurrence of need not imply occurrence of , since need not be

Example 1.0.5 The events , of the sample space in Example 1.0.1 are mutually exclusive.

But the events , are not mutually exclusive.

Now we introduce the concept of probability of events (in other words probability measure). Intuitivelyprobability quantifies the chance of the occurrence of an event. We say that an event has occurred, if theoutcome belongs to the event. In general it is not possible to assign probabilities to all events from thesample space. For the experiment given in Example 1.0.3, it is not possible to assign probabilities to all

subsets of . So one need to restrict to a smaller class of subsets of the sample space. For the

andom experiment given in Example 1.0.3, it turns out that one can assign probability to each interval inas its length. Therefore, one can assign probability to any finite union of intervals in , by

epresentating the finite union of intervals as a finite disjoint union of intervals. In fact one can assign

probability to any countable union interval in by preserving the desirable property "probability of

countable disjoint union is the sum of probabilities". Also note that if one can assign probability to anevent, then one can assign probability to its compliment, since occurence of the event is same as thenon-occurance of its compliemt. Thus one seek to define probability on those class of events whichatisfies "closed under complimentation" and "closed under countable union". This leads to the followingpecial family of events where one can assign probabilities.

Definition 1.4 A family of subsets of a nonempty set is said to be a -field if it satisfies theollowing.

i)


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ii) if , then

iii) if , then .

Example 1.0.6 Let be a nonempty set. Define

Then are -fields. Moreover, if is a -field of subsets of , then

.e., is the smallest and is the largest -field of subsets of .

Example 1.0.7 Let be a nonempty set and . Define

Then is a -field and is the smallest -field containing the set .

is called the -field generated by .

Lemma 1.0.1 Let be an index set and be a family of -fields. Then

s a -field.

Proof. Since for all , we have . Now,

Similarly it follows that

Hence is a -field.

Example 1.0.8 Let . Then

s a -field and is the smallest -field containing . We denote it by

This can be seen as follows. From Lemma 1.0.1, is a -field. From the definition of ,

clearly . If is a -field containing , then . Hence, is the smallest

-field containing .

Definition 1.5 A family of subsets of a non empty set is said to be a field if

i)

ii) if , then


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iii) if , then .

Example 1.0.9 Any -field is a field. In particular, are fields.

Example 1.0.10 Let . Define

Then is a field but not a -field.

Note that (i) and (ii) in the definition of field follows easily. To see (iii), for , if both

are finite so is and if either or is finite, then is finite. Hence

iii) follows. i.e., is a field.

To see that is not a -field, take

Now

Definition 1.6 (Probability measure)Let be a nonempty set and be a -field of subsets of . A map is said to be

a probability measure if P satisfies

i)

ii) if are pairwise disjoint, then

Definition 1.7 (Probability space).The triplet ; where , a nonempty set (sample space), , a -field and , a probability

measure; is called a probability space.

Example 1.0.11 Let . Define on as follows.

Then is a probability space. This probability space corresponds to the random experiment of

ossing an unbiased coin and noting the face.

Example 1.0.12 Let . Define on as follows.

Then is a probability space.

Solution.


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f are pairwise disjoint. Then

This holds since 's are disjoint)

Therefore

Therefore properties (i) , (ii) are satisfied. Hence is a probability measure.

Theorem 1.0.1 (Properties of probability measure) Let a probability space and

are in . Then

1) .

2) Finite sub-additivity:

3)Monotonicity: if , then

4)Boole's inequality (Countable sub-additivity):

5)Inclusion - exclusion formula:

6)Continuity property:

i) For

ii) For ,

Proof. Since


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This proves (1). Now

Therefore

since . This proves (3).

We prove (5) by induction. For

and

Here Hence we have

and

Combining the above, we have

Assume that equality holds for Consider


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Therefore the result true for . Hence by induction property (5) follows.

From property (5), we have

Hence

Thus we have (2).

Now we prove (6)(i). Set

Then

are disjoint and

(1.0.1)

Also

(1.0.2)


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Using (1.0.1), we get

Now using the definition of convergence of series, one has

(1.0.3)

Hence from (1.0.3), we have

Proof of (6)(ii) is as follows.

Note that

Now using (6)(i) we have

.e.,

Hence

From property (2), it follows that

.e.,

Therefore

(1.0.4)

Set

Then and are in . Also

Hence

(1.0.5)

Here the second equality follows from the continuity property 6(i). Using (1.0.5), letting in1.0.4), we have


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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. Now let us look at a random experiment

with uncountable sample space and the -field as a proper subset of the power set.

Consider the random experiment in Example 1.0.3, i.e, pick a point 'at random' from the interval .

Since point is picked 'at random', the probability measure should satisfy the following.

(1.0.6)

The -field we are using to define P is , the -field generated by all intervals in .

s called the Borel -field of subsets of .

Our aim is to define for all elements of , preserving (1.0.6). Set

Clearly .

Let . then can be represented as

where ,

Then

where

Therefore .

For , it follows from the definition of that .

Hence is a field.

Define on as follows.

(1.0.7)

where

's are pair wise disjoint intervals of the form .

Extension of from to follows from the extension theorem by Caratheodary. To understand the

statement of the extension theorem, we need the following definition.

Definition 1.8 (Probability measure on a field) Let be a nonempty set and be a field. Then


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is said to be a probability measure on if

i)

ii) if be such that are pairwise disjoint and , then

Example 1.0.13 The set function given by (1.0.7) is a probability measure on the field .

Theorem 1.0.2 (Extension Theorem) A probability measure defined on a field has a unique

extension to .

Using Theorem 1.0.2, one can extend defined by (1.0.7) to . Since

see Exercise 1.6 , there exists a unique probability measure on preserving (1.0.6).

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Maths Probability lec1/8.pdf