{"id":436,"date":"2018-12-21T04:28:06","date_gmt":"2018-12-21T04:28:06","guid":{"rendered":"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=436"},"modified":"2018-12-21T04:28:37","modified_gmt":"2018-12-21T04:28:37","slug":"characteristic-functions-v","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/chapter\/characteristic-functions-v\/","title":{"rendered":"Characteristic Functions V"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/qYBwzrzRj6E\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n<img class=\"aligncenter size-full wp-image-437\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-01.jpg\" alt=\"\" width=\"1345\" height=\"1721\" \/> <img class=\"aligncenter size-full wp-image-438\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-02.jpg\" alt=\"\" width=\"1225\" height=\"1541\" \/> <img class=\"aligncenter size-full wp-image-439\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-03.jpg\" alt=\"\" width=\"1297\" height=\"1633\" \/> <img class=\"aligncenter size-full wp-image-440\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-04.jpg\" alt=\"\" width=\"1289\" height=\"1593\" \/> <img class=\"aligncenter size-full wp-image-441\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-05.jpg\" alt=\"\" width=\"1265\" height=\"1617\" \/> <img class=\"aligncenter size-full wp-image-442\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-06.jpg\" alt=\"\" width=\"1265\" height=\"1569\" \/> <img class=\"aligncenter size-full wp-image-443\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-07.jpg\" alt=\"\" width=\"1257\" height=\"1633\" \/> <img class=\"aligncenter size-full wp-image-444\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-08.jpg\" alt=\"\" width=\"1305\" height=\"1617\" \/> <img class=\"aligncenter size-full wp-image-445\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-09.jpg\" alt=\"\" width=\"1337\" height=\"1601\" \/> <img class=\"aligncenter size-full wp-image-446\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-10.jpg\" alt=\"\" width=\"1253\" height=\"1609\" \/> <img class=\"aligncenter size-full wp-image-447\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-11.jpg\" alt=\"\" width=\"1281\" height=\"1585\" \/> <img class=\"aligncenter size-full wp-image-448\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12.jpg\" alt=\"\" width=\"1289\" height=\"1609\" \/> <img class=\"aligncenter size-full wp-image-449\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13.jpg\" alt=\"\" width=\"1345\" height=\"1609\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Characteristic Functions V <\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/qYBwzrzRj6E\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nReferences\r\n<ol>\r\n \t<li>TM Apostol: Mathematical Analysis.<\/li>\r\n \t<li>AK Basu: Measure Theory and Probability.<\/li>\r\n \t<li>P Billingsley: Probability and Measure.<\/li>\r\n \t<li>KL Chung: A Course in Probability Theory.<\/li>\r\n \t<li>W Feller: An Introduction to Probability Theory and its Applications, Vol-I.<\/li>\r\n \t<li>AM Gun, MK Gupta, B Dasgupta: An Outline of Statistical Theory, Vol -I.<\/li>\r\n \t<li>E Luckacs, RG Laha: Applications of Characteristic Functions.<\/li>\r\n \t<li>E Parzen: Modern Probability Theory and its Applications.<\/li>\r\n \t<li>RJ Ser ing: Approximation Theorems of Mathematical Statistics.<\/li>\r\n \t<li>W Rudin: Principles of Mathematical Analysis<\/li>\r\n<\/ol>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/qYBwzrzRj6E\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-437\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-01.jpg\" alt=\"\" width=\"1345\" height=\"1721\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-01.jpg 1345w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-01-234x300.jpg 234w, 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350w\" sizes=\"auto, (max-width: 1281px) 100vw, 1281px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-448\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12.jpg\" alt=\"\" width=\"1289\" height=\"1609\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12.jpg 1289w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12-240x300.jpg 240w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12-768x959.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12-820x1024.jpg 820w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12-65x81.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12-225x281.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-12-350x437.jpg 350w\" sizes=\"auto, (max-width: 1289px) 100vw, 1289px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-449\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13.jpg\" alt=\"\" width=\"1345\" height=\"1609\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13.jpg 1345w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13-251x300.jpg 251w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13-768x919.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13-856x1024.jpg 856w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13-65x78.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13-225x269.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774313CharacteristicFunctions-5-13-350x419.jpg 350w\" sizes=\"auto, (max-width: 1345px) 100vw, 1345px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Characteristic Functions V <\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/qYBwzrzRj6E\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>References<\/p>\n<ol>\n<li>TM Apostol: Mathematical Analysis.<\/li>\n<li>AK Basu: Measure Theory and Probability.<\/li>\n<li>P Billingsley: Probability and Measure.<\/li>\n<li>KL Chung: A Course in Probability Theory.<\/li>\n<li>W Feller: An Introduction to Probability Theory and its Applications, Vol-I.<\/li>\n<li>AM Gun, MK Gupta, B Dasgupta: An Outline of Statistical Theory, Vol -I.<\/li>\n<li>E Luckacs, RG Laha: Applications of Characteristic Functions.<\/li>\n<li>E Parzen: Modern Probability Theory and its Applications.<\/li>\n<li>RJ Ser ing: Approximation Theorems of Mathematical Statistics.<\/li>\n<li>W Rudin: Principles of Mathematical Analysis<\/li>\n<\/ol>\n","protected":false},"author":3,"menu_order":30,"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-436","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\/436","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":1,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/436\/revisions"}],"predecessor-version":[{"id":450,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/436\/revisions\/450"}],"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\/436\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/media?parent=436"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapter-type?post=436"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/contributor?post=436"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/license?post=436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}