{"id":399,"date":"2018-12-21T04:07:58","date_gmt":"2018-12-21T04:07:58","guid":{"rendered":"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=399"},"modified":"2018-12-21T04:16:54","modified_gmt":"2018-12-21T04:16:54","slug":"399","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/chapter\/399\/","title":{"rendered":"Characteristic Functions III"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/PFy_jRyYv7c\" 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-410\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1.jpg\" alt=\"\" width=\"1335\" height=\"1779\" \/> <img class=\"aligncenter size-full wp-image-411\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1.jpg\" alt=\"\" width=\"1345\" height=\"1641\" \/> <img class=\"aligncenter size-full wp-image-412\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1.jpg\" alt=\"\" width=\"1377\" height=\"1633\" \/>\r\n\r\n<img class=\"aligncenter size-full wp-image-419\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2.jpg\" alt=\"\" width=\"1337\" height=\"1745\" \/> <img class=\"aligncenter size-full wp-image-420\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2.jpg\" alt=\"\" width=\"1365\" height=\"1601\" \/> <img class=\"aligncenter size-full wp-image-421\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2.jpg\" alt=\"\" width=\"1217\" height=\"1641\" \/> <img class=\"aligncenter size-full wp-image-422\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2.jpg\" alt=\"\" width=\"1345\" height=\"1657\" \/> <img class=\"aligncenter size-full wp-image-423\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2.jpg\" alt=\"\" width=\"1369\" height=\"857\" \/>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Characteristic Functions III <\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/PFy_jRyYv7c\" 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\/PFy_jRyYv7c\" 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-410\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1.jpg\" alt=\"\" width=\"1335\" height=\"1779\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1.jpg 1335w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1-225x300.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1-768x1024.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1-65x87.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-1-1-350x466.jpg 350w\" sizes=\"auto, (max-width: 1335px) 100vw, 1335px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-411\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1.jpg\" alt=\"\" width=\"1345\" height=\"1641\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1.jpg 1345w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1-246x300.jpg 246w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1-768x937.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1-839x1024.jpg 839w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1-65x79.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1-225x275.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-2-1-350x427.jpg 350w\" sizes=\"auto, (max-width: 1345px) 100vw, 1345px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-412\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1.jpg\" alt=\"\" width=\"1377\" height=\"1633\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1.jpg 1377w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1-253x300.jpg 253w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1-768x911.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1-863x1024.jpg 863w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1-65x77.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1-225x267.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-3-1-350x415.jpg 350w\" sizes=\"auto, (max-width: 1377px) 100vw, 1377px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-419\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2.jpg\" alt=\"\" width=\"1337\" height=\"1745\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2.jpg 1337w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2-230x300.jpg 230w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2-768x1002.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2-785x1024.jpg 785w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2-65x85.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2-225x294.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-4-2-350x457.jpg 350w\" sizes=\"auto, (max-width: 1337px) 100vw, 1337px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-420\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2.jpg\" alt=\"\" width=\"1365\" height=\"1601\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2.jpg 1365w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2-256x300.jpg 256w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2-768x901.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2-873x1024.jpg 873w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2-65x76.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2-225x264.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-5-2-350x411.jpg 350w\" sizes=\"auto, (max-width: 1365px) 100vw, 1365px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-421\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2.jpg\" alt=\"\" width=\"1217\" height=\"1641\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2.jpg 1217w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2-222x300.jpg 222w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2-768x1036.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2-759x1024.jpg 759w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2-65x88.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2-225x303.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-6-2-350x472.jpg 350w\" sizes=\"auto, (max-width: 1217px) 100vw, 1217px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-422\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2.jpg\" alt=\"\" width=\"1345\" height=\"1657\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2.jpg 1345w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2-244x300.jpg 244w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2-768x946.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2-831x1024.jpg 831w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2-65x80.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2-225x277.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-7-2-350x431.jpg 350w\" sizes=\"auto, (max-width: 1345px) 100vw, 1345px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-423\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2.jpg\" alt=\"\" width=\"1369\" height=\"857\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2.jpg 1369w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2-300x188.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2-768x481.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2-1024x641.jpg 1024w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2-65x41.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2-225x141.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774233CharacteristicFunctions-3-8-2-350x219.jpg 350w\" sizes=\"auto, (max-width: 1369px) 100vw, 1369px\" \/><\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Characteristic Functions III <\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/PFy_jRyYv7c\" 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":28,"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-399","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\/399","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":7,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/399\/revisions"}],"predecessor-version":[{"id":427,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/399\/revisions\/427"}],"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\/399\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/media?parent=399"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapter-type?post=399"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/contributor?post=399"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/license?post=399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}