Maths Probability Random Variables lec2/8.pdf

7/27/2019 Maths Probability Random Variables lec2/8.pdf

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Chapter 2

Random Variables

Lectures 4 - 7

In many situations, one is interested in only some aspects of the random experiment. For example,

consider the experiment of tossing unbiased coins and we are only interested in number of 'Heads'

turned up. The probability space corresponding to the experiment of tossing coins is given by

and is described by

Our interest is in noting , i.e., in the map . So our

nterest is only on certain function of the sample space. But all functions defined on the sample spaceare not useful, in the sense that we may not be able to assign probabilities to all basic events associatedwith the function. So one need to restrict to certain class of functions of the sample space. This

motivates us to define random variables.

Definition 2.1 Let be a probability space. A function is said to be a

random variable if

Now on, we denote by .

Example 2.0.14 Let . Define as follows:

Then is a random variable.

Example 2.0.15 Let . Define .

For

Since

is a random variable.


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Note that any map is a random variable for the probability space given in Example 2.0.14

but this is not the case with probability space given in Example 2.0.15.

is a random variable imply that the basic events associated are in . Does

this imply a larger class of events associated with can be assigned probabilities (i.e. in ) ? To

examine this, one need the following -field.

Definition 2.2 The -field generated by the collection of all open sets in is called the Borel -field

of subsets of and is denoted by .

Lemma 2.0.2

Let . Then

Proof. For ,

Therefore

Hence

For ,

Therefore

Also, for

Therefore, contains all open intervals. Since any open set in can be written as a countable

union of open intervals, it follows that

where denote the set of all open sets in . Thus,

This completes the proof.

Theorem 2.0.3 Let be a function. Then is a random variable on iff


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for all .

Proof. Suppose is a random variable. i.e.,

(2.0.1)

Set

From (2.0.1), we have

(2.0.2)

Note that . Hence .

Now

Also

Hence is a -field. Now from (2.0.2) and Lemma 2.0.1, it follows that . i.e,

for all .

Converse statement follows from the observation


Remark 2.0.1 Theorem 2.0.3 tells that if for all , then

for all .

Theorem 2.0.4 Let be a random variable. Define

Then is a -field.

Proof.


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Hence .

For , there exists such that

and

Hence implies . Similarly from

t follows that


Definition 2.3 The sigma field is called the sigma field generated by .

Remark 2.0.2 The occurrence or nonoccurence of the event tells whether any realization

is in or not. Thus collects all such information. Hence can be viewed as the

nformation available with the random variable.

Example 2.0.16 Let . Then

Theorem 2.0.5 If are random variables on and , then

and are random variables.

Proof: The proof of first two are simple exercises.

For ,

[Here means ] Now

Hence for all . Therefore is a random variable.


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For ,

Hence, is a random variable.

Note that

Since are random variables and is their sum, is a random

variable.

Theorem 2.0.6

(i) If are random variables, so are , .

(ii) If is a sequence of random variables and

Then is a random variable. Analogous statement is true for infimum.

Proof:

(i) Set . For ,

Therefore is a random variable.

The proof of is a random variable is similar.

Proof of (ii) follows from

Theorem 2.0.7 Let be a sequence of random variables on such that

Then defined by

s a random variable.

Proof. For .


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Hence

Now suppose

If , then there exists there exists and such that

(2.0.3)

(This follows, since )

Also

This contradicts (2.0.3). Therefore

Hence we have

(2.0.4)

Since and is a -field, using(2.0.4) it follows from the definition of -field

that

Therefore is a random variable.

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Maths Probability Random Variables lec2/8.pdf