Maths Conditional Probability and Independence lec3/8.pdf

7/27/2019 Maths Conditional Probability and Independence lec3/8.pdf

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Chapter 3

Conditional Probability and Independence

ectures 8 -12

n this chapter, we introduce the concepts of conditional probability and independence.

uppose we know that an event already occurred. Then a natural question asked is ``What is the effectf this on the probabilities of other events?'' This leads to the notion of conditional probability.

Definition 3.1 Let be a probability space and be such that The

onditional probability of an event given denoted by is defined as

efine

hen is a -field of subsets of (see Exercise 3.2).

collection of events is said to be a partition of , if

) 's are pairwise disjoint

i)

ere may be . If , then partition is said to be finite partition and if , it is calledcountable partition.

Theorem 3.0.8 Define on as follows.

hen is a probability space.

roof is an exercise, see Exercise 3.3.

Theorem 3.0.9 (Law of total probability-discrete form) Let be a probability space and

be a partition of such that for all . Then for ,



roof.

Theorem 3.0.10 (Bayes Theorem) Let be a probability space and such that

and . Then

roof.

ow use Law of total probability to complete the proof.

Definition 3.2 (Independence ) Two events are said to be independent if

Remark 3.0.3 If , then and are independent iff This confirms

he intuition behind the notion of independence, ``the occurrence one event doesn't have any effect on theccurrence of the other''.

xample 3.0.17 Define the probability space as follows.

nd

onsider the events and . Then and

are independent and and are dependent.

xample 3.0.18 Consider the probability space defined by



nd given by (1.0.6). Consider the events . Then

are independent and are dependent.

he notion of independence defined above can be extended to independence of three or more events in theollowing manner.

Definition 3.3 (Independence of three events). The events are independent (mutually) if

) ; and are independent and

i)

f the events satisfies only (i), then are said to be pairwise independent.

xample 3.0.19 Define and

onsider the events

hen are pairwise independent but not independent.

xample 3.0.20 Define as follows.

nd

et

hen are independent.

Definition 3.4 The events are said to be independent if for any distinct

from

sing the notion of independence of events, one can define the independence of -fields as follows.



Definition 3.5 Two -fields of subsets of are said to be independent if for any

Definition 3.6 The -fields of subsets of are said to be independent if

are independent whenever .

ne can go a step further to define independence of any family of -fields as follows.

Definition 3.7 A family of -fields , where is an index set, are independent if for any

nite subset , the -fields are independent.

inally one can introduce the notion of independence of random variables through the corresponding -

eld generated. It is natural to define independence of random variables using the corresponding -fields,

nce the -field contains all information about the random variable.

Definition 3.8 Two random variables and are independent if and are

ndependent.

Definition 3.9 A family of random variables are independent if are

ndependent.

xample 3.0.21 Let are independent events iff and are independent -fields iff

he random variables and are independent.

r o o f . Let and are independent. Consider

ence and are independent. Changing the roles of and we have and are

ndependent. Now and are independent implies that and are independent.

ince

follows that and are independent. Converse statement is obvious.

he second part follows from



xample 3.0.22 The trivial -field is independent of any -field of subsets of .

xample 3.0.23 Define a probability space as follows:

nd

or set

hen are independent.

his can be seen from the following.

ote that

ence

lso

xample 3.0.24 Let is given by as

nd



efine two random variables as

hen and are independent. Here note that

nd

We conclude this chapter with the celebrated Borel-Cantelli Lemma. To this end, we need the notion of msup and liminf of events.

Definition 3.10 of sets) For subsets of , define

imilarly

n the following theorem, we list some useful properties of limsup and liminf. This will make the objects

and of sets much clearer.

heorem 3.0.11

.Let

hen

.Let

hen



.The following inclusion holds.

.The following identity holds.

.If , then

.If then

roof.

.Consider

his proves (1).

.Consider

hus (2).

.Proof of (3) follows from (1) and (2).

.Proof of (4) follows from De Morgen's laws.

.Consider

imilarly

his proves (5)



.The proof of (6) follows from (4) and (5).

Remark 3.0.4 The properties (1)-(6) are analogous to the corresponding properties of and

of real numbers.

Remark 3.0.5 Analogous to the notion of limit of sequence of numbers, one can say thatexists if

ence from Theorem 3.0.11 (5), if is an increasing sequence of sets, then exists

nd is . Similarly using Theorem 3.0.11 (6), if is an decreasing sequence of sets, then

exists and is . Now student can see why property (6) in Theorem 1.0.1 is called

ontinuity property of probability.

Theorem 3.0.12 (Borel - Cantelli Lemma) Let be a probability space and

.

) If

hen

i) If

nd

re independent, then

Proof: (i) Consider



ote that the r.h.s as , since

herefore

i) Note that

herefore

ow

he last equality follows from

sing , we get

ince

we get

herefore



hus

herefore

his completes the proof.

xample 3.0.25 Define a probability space as follows.

et

where

et

hen is a field, see Exercise 3.5. Define

or , define

or , there exists which are pairwise disjoint such that

efine as follows.

ow extend to using the extension theorem.

et

ne can calculate using Borel - Cantelli Lemma as follows. Define



hen

re independent and

ence

lso note that

herefore using Borel - Cantelli Lemma

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Maths Conditional Probability and Independence lec3/8.pdf