{"id":428,"date":"2018-12-21T04:20:53","date_gmt":"2018-12-21T04:20:53","guid":{"rendered":"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=428"},"modified":"2018-12-21T04:21:11","modified_gmt":"2018-12-21T04:21:11","slug":"characteristic-functions-iv","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/chapter\/characteristic-functions-iv\/","title":{"rendered":"Characteristic Functions IV"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/3fxREh7YP4s\" 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-429\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1.jpg\" alt=\"\" width=\"1261\" height=\"1529\" \/> <img class=\"aligncenter size-full wp-image-430\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2.jpg\" alt=\"\" width=\"1233\" height=\"1565\" \/> <img class=\"aligncenter size-full wp-image-431\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3.jpg\" alt=\"\" width=\"1237\" height=\"1601\" \/> <img class=\"aligncenter size-full wp-image-432\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4.jpg\" alt=\"\" width=\"1313\" height=\"1545\" \/> <img class=\"aligncenter size-full wp-image-433\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5.jpg\" alt=\"\" width=\"1265\" height=\"1665\" \/> <img class=\"aligncenter size-full wp-image-434\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6.jpg\" alt=\"\" width=\"1393\" height=\"985\" \/>\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 IV <\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/3fxREh7YP4s\" 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>\r\n&nbsp;","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/3fxREh7YP4s\" 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-429\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1.jpg\" alt=\"\" width=\"1261\" height=\"1529\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1.jpg 1261w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1-247x300.jpg 247w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1-768x931.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1-845x1024.jpg 845w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1-65x79.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1-225x273.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-1-350x424.jpg 350w\" sizes=\"auto, (max-width: 1261px) 100vw, 1261px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-430\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2.jpg\" alt=\"\" width=\"1233\" height=\"1565\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2.jpg 1233w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2-236x300.jpg 236w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2-768x975.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2-807x1024.jpg 807w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2-65x83.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2-225x286.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-2-350x444.jpg 350w\" sizes=\"auto, (max-width: 1233px) 100vw, 1233px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-431\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3.jpg\" alt=\"\" width=\"1237\" height=\"1601\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3.jpg 1237w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3-232x300.jpg 232w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3-768x994.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3-791x1024.jpg 791w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3-65x84.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3-225x291.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-3-350x453.jpg 350w\" sizes=\"auto, (max-width: 1237px) 100vw, 1237px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-432\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4.jpg\" alt=\"\" width=\"1313\" height=\"1545\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4.jpg 1313w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4-255x300.jpg 255w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4-768x904.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4-870x1024.jpg 870w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4-65x76.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4-225x265.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-4-350x412.jpg 350w\" sizes=\"auto, (max-width: 1313px) 100vw, 1313px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-433\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5.jpg\" alt=\"\" width=\"1265\" height=\"1665\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5.jpg 1265w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5-228x300.jpg 228w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5-768x1011.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5-778x1024.jpg 778w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5-65x86.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5-225x296.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-5-350x461.jpg 350w\" sizes=\"auto, (max-width: 1265px) 100vw, 1265px\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-434\" src=\"http:\/\/statp01.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6.jpg\" alt=\"\" width=\"1393\" height=\"985\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6.jpg 1393w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6-300x212.jpg 300w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6-768x543.jpg 768w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6-1024x724.jpg 1024w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6-65x46.jpg 65w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6-225x159.jpg 225w, https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-content\/uploads\/sites\/125\/2018\/12\/1525774285CharacteristicFunctions-4-6-350x247.jpg 350w\" sizes=\"auto, (max-width: 1393px) 100vw, 1393px\" \/><\/p>\n<p>&nbsp;<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Characteristic Functions IV <\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/3fxREh7YP4s\" 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<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":29,"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-428","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\/428","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\/428\/revisions"}],"predecessor-version":[{"id":435,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapters\/428\/revisions\/435"}],"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\/428\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/media?parent=428"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/pressbooks\/v2\/chapter-type?post=428"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/contributor?post=428"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/statp01\/wp-json\/wp\/v2\/license?post=428"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}