{"id":278,"date":"2018-12-20T06:32:37","date_gmt":"2018-12-20T06:32:37","guid":{"rendered":"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=278"},"modified":"2018-12-20T06:32:51","modified_gmt":"2018-12-20T06:32:51","slug":"random-variable-and-measurable-transformation","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/chapter\/random-variable-and-measurable-transformation\/","title":{"rendered":"Random Variable and Measurable Transformation"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n \r\n\r\n \r\n\r\n \r\n

Statistics is all about randomness. So the one concept which lies at the heart of statistics is random variable. As the name suggests \u2019random\u2019 and \u2019variable\u2019, here \u2019variable\u2019 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 \u2019random\u2019. So in this module we try to get a feel of the concept of \u2019random variable\u2019. We start by defining random variable.<\/p>\r\n \r\n

\u00a0Definition of Random Variable<\/strong><\/p>\r\n

In many cases we shall find the elements of \u2126,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 \u2126 t1, 2,..., 6u. Even when the elementary events\u00a0 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 \u2126 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 \u2126 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 p\u2126,F,Pq we mean a finite real valued function, say X such that\r\nt\u03c9 : Xp\u03c9q \u00a4 xu P F @x P R\r\n1\r\nExample 1 A coin is tossed twice. Denoting the occurence of \u2019<\/p>\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\r\n\r\n
you can view video on -Random Variable and Measurable Transformation<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nReferences\r\n\r\n1. TM Apostol: Mathematical Analysis.\r\n2. AK Basu: Measure Theory and Probability.\r\n3. P Billingsley: Probability and Measure.\r\n4. KL Chung: A Course in Probability Theory.\r\n5. W Feller: An Introduction to Probability Theory and its Applications, Vol-I.\r\n6. AM Gun, MK Gupta, B Dasgupta: An Outline of Statistical Theory, Vol -I.\r\n7. E Luckacs, RG Laha: Applications of Characteristic Functions.\r\n8. E Parzen: Modern Probability Theory and its Applications.\r\n9. RJ Serfling: Approximation Theorems of Mathematical Statistics.\r\n10. W Rudin: Principles of Mathematical Analy","rendered":"
\"epgp<\/a>
\n<\/span><\/div>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

Statistics is all about randomness. So the one concept which lies at the heart of statistics is random variable. As the name suggests \u2019random\u2019 and \u2019variable\u2019, here \u2019variable\u2019 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 \u2019random\u2019. So in this module we try to get a feel of the concept of \u2019random variable\u2019. We start by defining random variable.<\/p>\n

 <\/p>\n

\u00a0Definition of Random Variable<\/strong><\/p>\n

In many cases we shall find the elements of \u2126,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 \u2126 t1, 2,…, 6u. Even when the elementary events\u00a0 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 \u2126 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 \u2126 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 p\u2126,F,Pq we mean a finite real valued function, say X such that
\nt\u03c9 : Xp\u03c9q \u00a4 xu P F @x P R
\n1
\nExample 1 A coin is tossed twice. Denoting the occurence of \u2019<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n\n\n\n
you can view video on -Random Variable and Measurable Transformation<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

References<\/p>\n

1. TM Apostol: Mathematical Analysis.
\n2. AK Basu: Measure Theory and Probability.
\n3. P Billingsley: Probability and Measure.
\n4. KL Chung: A Course in Probability Theory.
\n5. W Feller: An Introduction to Probability Theory and its Applications, Vol-I.
\n6. AM Gun, MK Gupta, B Dasgupta: An Outline of Statistical Theory, Vol -I.
\n7. E Luckacs, RG Laha: Applications of Characteristic Functions.
\n8. E Parzen: Modern Probability Theory and its Applications.
\n9. RJ Serfling: Approximation Theorems of Mathematical Statistics.
\n10. W Rudin: Principles of Mathematical Analy<\/p>\n","protected":false},"author":3,"menu_order":15,"template":"","meta":{"_acf_changed":false,"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["mr-samopriya-basu"],"pb_section_license":""},"chapter-type":[],"contributor":[59],"license":[],"class_list":["post-278","chapter","type-chapter","status-publish","hentry","contributor-mr-samopriya-basu"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/278","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":2,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/278\/revisions"}],"predecessor-version":[{"id":287,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/278\/revisions\/287"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/278\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/media?parent=278"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapter-type?post=278"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/contributor?post=278"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/license?post=278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}