15 Random Variable and Measurable Transformation

Mr. Samopriya Basu

Statistics is all about randomness. So the one concept which lies at the heart of statistics is random variable. As the name suggests ’random’ and ’variable’, here ’variable’ means it will vary, it will take some values, it will belong to some intervals with some law or rule that is governed by this process ’random’. So in this module we try to get a feel of the concept of ’random variable’. We start by defining random variable.

 Definition of Random Variable

In many cases we shall find the elements of Ω,i.e. the elementary events are themselves expressed as real numbers. For example while throwing a die once any of the numbers 1, 2,…, 6 can appear. Therefore the sample space is Ω t1, 2,…, 6u. Even when the elementary events  themselves are not numbered we can associate some definite real number to each elementary events according to some given rule. In tossing a coin once the sample space Ω is tH,T u. If we assign the number 1 to H and 0 to T then the sample space can also be written as Ω t0, 1u. Often when an experiment is performed our main interest lies in the function of the outcome rather than the actual outcome itself. For example, in a dice throwing game a gambler is more interested in his gains rather than his number of wins and loses. It is therefore desirable to introduce a point function on the sample space. these quantities of interest or more formally these real valued functions defined on the sample space are known as random variables. We give here the formal definition of random variable: Definition 1 By a random variable with respect to the probability space pΩ,F,Pq we mean a finite real valued function, say X such that
tω : Xpωq ¤ xu P F @x P R
1
Example 1 A coin is tossed twice. Denoting the occurence of ’

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