{"id":3055,"date":"2019-12-15T20:39:55","date_gmt":"2019-12-15T20:39:55","guid":{"rendered":"https:\/\/nenadnoveljic.com\/blog\/?p=3055"},"modified":"2019-12-17T22:12:26","modified_gmt":"2019-12-17T22:12:26","slug":"expected-payoff-of-the-gaussian-distribution-tail","status":"publish","type":"post","link":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/","title":{"rendered":"Expected Payoff of the Gaussian Distribution Tail"},"content":{"rendered":"<p>I wrote this note primarily for myself to understand the equation (6) on the page 5 from <a href=\"https:\/\/en.wikipedia.org\/wiki\/Nassim_Nicholas_Taleb\">Nassim Nicholas Taleb&#8217;s<\/a> paper <a href=\"https:\/\/arxiv.org\/pdf\/1907.11162.pdf\">On the Statistical Differences between Binary Forecasts and Real World Payoffs<\/a>:<\/p>\n<p><a name=\"id215278451\"><\/a><\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 47px;\"><span class=\"ql-right-eqno\"> (1) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-60bcb1a08b68bc794bd35fe6f44cfe16_l3.png\" height=\"47\" width=\"166\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#32; &#92;&#105;&#110;&#116;&#95;&#123;&#75;&#125;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#120;&#103;&#40;&#120;&#41;&#100;&#120;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#101;&#94;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#75;&#94;&#50;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#125; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>where:<\/p>\n<ul>\n<li><i>x<\/i> is a random variable,<\/li>\n<li><i>g(x)<\/i> is the probability density function (PDF) of <i>x<\/i>, which is <a href=\"https:\/\/en.wikipedia.org\/wiki\/Normal_distribution\">Gaussian distribution<\/a> (centered and normalized).<\/li>\n<\/ul>\n<p>The author assumes familiarity with the equation <a href=\"#id215278451\">1<\/a> and, therefore, doesn&#8217;t elaborate on it. The steps below show where this equation came from.<\/p>\n<p>More generally, the expected payoff <i>I<\/i> for all values of <i>x<\/i> above <i>K<\/i>, i.e. the tail of the distribution, is:<\/p>\n<p><a name=\"id767238686\"><\/a><\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 41px;\"><span class=\"ql-right-eqno\"> (2) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-9c2461482ecd6b1c6f7de3c149f1d0e4_l3.png\" height=\"41\" width=\"155\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#32; &#73;&#32;&#61;&#32;&#92;&#105;&#110;&#116;&#95;&#123;&#75;&#125;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#102;&#40;&#120;&#41;&#103;&#40;&#120;&#41;&#100;&#120; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>where <i>f(x)<\/i> is the payoff function.<\/p>\n<p>Normally, we would use <a href=\"https:\/\/en.wikipedia.org\/wiki\/Integration_by_parts\">integration by parts<\/a> for calculating the integral above.<\/p>\n<p>But the author considers a special case where the payoff function <i>f(x)<\/i> is linear:<br \/>\n<a name=\"id1771414392\"><\/a><\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 18px;\"><span class=\"ql-right-eqno\"> (3) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-ee85d5bef6000e00089e935e734463f1_l3.png\" height=\"18\" width=\"68\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#32; &#102;&#40;&#120;&#41;&#61;&#120; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>So, <a href=\"#id767238686\">2<\/a> becomes:<br \/>\n<a name=\"id3611251395\"><\/a><\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 41px;\"><span class=\"ql-right-eqno\"> (4) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-c20b6fd225bff17ba72d1a7eacc764cb_l3.png\" height=\"41\" width=\"130\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#32; &#73;&#32;&#61;&#32;&#92;&#105;&#110;&#116;&#95;&#123;&#75;&#125;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#120;&#103;&#40;&#120;&#41;&#100;&#120; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>The integral <a href=\"#id3611251395\">4<\/a> is much easier to solve than more general <a href=\"#id767238686\">2<\/a>, because the first derivative of the centered and normalized Gaussian PDF has the following property:<\/p>\n<p><a name=\"id2221150966\"><\/a><\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 20px;\"><span class=\"ql-right-eqno\"> (5) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-7d0cf3af214d80b7aaf1dbed134f8ebf_l3.png\" height=\"20\" width=\"117\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;&#32; &#103;&#39;&#40;&#120;&#41;&#32;&#61;&#32;&#45;&#32;&#120;&#32;&#103;&#40;&#120;&#41; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>We will integrate the equation and apply the second <a href=\"https:\/\/en.wikipedia.org\/wiki\/Fundamental_theorem_of_calculus\">fundamental theorem of calculus<\/a> to the left side of <a href=\"#id2221150966\">5<\/a>:<\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 41px;\"><span class=\"ql-right-eqno\"> (6) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-1e46b073eedb237866de80b23fc50a73_l3.png\" height=\"41\" width=\"270\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#108;&#105;&#109;&#95;&#123;&#120;&#32;&#92;&#116;&#111;&#32;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#32;&#103;&#40;&#120;&#41;&#32;&#45;&#32;&#103;&#40;&#75;&#41;&#61;&#32;&#45;&#32;&#92;&#105;&#110;&#116;&#95;&#123;&#75;&#125;&#94;&#123;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#32;&#120;&#32;&#103;&#40;&#120;&#41;&#32;&#100;&#120; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>Notice that the right side became our expected payoff function, therefore:<\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 25px;\"><span class=\"ql-right-eqno\"> (7) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-c1edd7bc4f45e55d7d3d183e72fbd09c_l3.png\" height=\"25\" width=\"165\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#103;&#40;&#75;&#41;&#32;&#45;&#32;&#92;&#108;&#105;&#109;&#95;&#123;&#120;&#32;&#92;&#116;&#111;&#32;&#92;&#105;&#110;&#102;&#116;&#121;&#125;&#103;&#40;&#120;&#41;&#32;&#61;&#32;&#73; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n<p>Since the probability approaches 0 as <i>x<\/i> approaches infinity, the expected payoff of all <i>x<\/i> above <i>K<\/i> equals the probability of <i>K<\/i>:<\/p>\n<p class=\"ql-left-displayed-equation\" style=\"line-height: 47px;\"><span class=\"ql-right-eqno\"> (8) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/nenadnoveljic.com\/blog\/wp-content\/ql-cache\/quicklatex.com-3c8113cfff21fb20037c9a13feb634bc_l3.png\" height=\"47\" width=\"140\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#73;&#32;&#61;&#32;&#103;&#40;&#75;&#41;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#101;&#94;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#75;&#94;&#50;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#125; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Expected payoff of the Gaussian distribution tail for a linear payoff function <a href=\"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/\" class=\"more-link\">Continue Reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[30],"tags":[],"class_list":["post-3055","post","type-post","status-publish","format-standard","hentry","category-math"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Expected Payoff of the Gaussian Distribution Tail - All-round Database Topics<\/title>\n<meta name=\"description\" content=\"Expected payoff of the Gaussian distribution tail for a linear payoff function\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Expected Payoff of the Gaussian Distribution Tail - All-round Database Topics\" \/>\n<meta property=\"og:description\" content=\"Expected payoff of the Gaussian distribution tail for a linear payoff function\" \/>\n<meta property=\"og:url\" content=\"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/\" \/>\n<meta property=\"og:site_name\" content=\"All-round Database Topics\" \/>\n<meta property=\"article:published_time\" content=\"2019-12-15T20:39:55+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2019-12-17T22:12:26+00:00\" \/>\n<meta name=\"author\" content=\"Nenad Noveljic\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@NenadNoveljic\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Nenad Noveljic\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/\"},\"author\":{\"name\":\"Nenad Noveljic\",\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/#\\\/schema\\\/person\\\/51458d9dd86dbbdd19f5add451d44efa\"},\"headline\":\"Expected Payoff of the Gaussian Distribution Tail\",\"datePublished\":\"2019-12-15T20:39:55+00:00\",\"dateModified\":\"2019-12-17T22:12:26+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/\"},\"wordCount\":357,\"commentCount\":0,\"articleSection\":[\"Math\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/\",\"url\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/\",\"name\":\"Expected Payoff of the Gaussian Distribution Tail - All-round Database Topics\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/#website\"},\"datePublished\":\"2019-12-15T20:39:55+00:00\",\"dateModified\":\"2019-12-17T22:12:26+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/#\\\/schema\\\/person\\\/51458d9dd86dbbdd19f5add451d44efa\"},\"description\":\"Expected payoff of the Gaussian distribution tail for a linear payoff function\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/expected-payoff-of-the-gaussian-distribution-tail\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Expected Payoff of the Gaussian Distribution Tail\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/#website\",\"url\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/\",\"name\":\"All-round Database Topics\",\"description\":\"Nenad Noveljic\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/#\\\/schema\\\/person\\\/51458d9dd86dbbdd19f5add451d44efa\",\"name\":\"Nenad Noveljic\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a97b796613ea48ec8a7b79c8ffe1c685dcffc920c68121f6238d5caab5070670?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a97b796613ea48ec8a7b79c8ffe1c685dcffc920c68121f6238d5caab5070670?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a97b796613ea48ec8a7b79c8ffe1c685dcffc920c68121f6238d5caab5070670?s=96&d=mm&r=g\",\"caption\":\"Nenad Noveljic\"},\"sameAs\":[\"nenad-noveljic-9b746a6\",\"https:\\\/\\\/x.com\\\/NenadNoveljic\"],\"url\":\"https:\\\/\\\/nenadnoveljic.com\\\/blog\\\/author\\\/nenad\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Expected Payoff of the Gaussian Distribution Tail - All-round Database Topics","description":"Expected payoff of the Gaussian distribution tail for a linear payoff function","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/","og_locale":"en_US","og_type":"article","og_title":"Expected Payoff of the Gaussian Distribution Tail - All-round Database Topics","og_description":"Expected payoff of the Gaussian distribution tail for a linear payoff function","og_url":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/","og_site_name":"All-round Database Topics","article_published_time":"2019-12-15T20:39:55+00:00","article_modified_time":"2019-12-17T22:12:26+00:00","author":"Nenad Noveljic","twitter_card":"summary_large_image","twitter_creator":"@NenadNoveljic","twitter_misc":{"Written by":"Nenad Noveljic","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/#article","isPartOf":{"@id":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/"},"author":{"name":"Nenad Noveljic","@id":"https:\/\/nenadnoveljic.com\/blog\/#\/schema\/person\/51458d9dd86dbbdd19f5add451d44efa"},"headline":"Expected Payoff of the Gaussian Distribution Tail","datePublished":"2019-12-15T20:39:55+00:00","dateModified":"2019-12-17T22:12:26+00:00","mainEntityOfPage":{"@id":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/"},"wordCount":357,"commentCount":0,"articleSection":["Math"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/","url":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/","name":"Expected Payoff of the Gaussian Distribution Tail - All-round Database Topics","isPartOf":{"@id":"https:\/\/nenadnoveljic.com\/blog\/#website"},"datePublished":"2019-12-15T20:39:55+00:00","dateModified":"2019-12-17T22:12:26+00:00","author":{"@id":"https:\/\/nenadnoveljic.com\/blog\/#\/schema\/person\/51458d9dd86dbbdd19f5add451d44efa"},"description":"Expected payoff of the Gaussian distribution tail for a linear payoff function","breadcrumb":{"@id":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/nenadnoveljic.com\/blog\/expected-payoff-of-the-gaussian-distribution-tail\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/nenadnoveljic.com\/blog\/"},{"@type":"ListItem","position":2,"name":"Expected Payoff of the Gaussian Distribution Tail"}]},{"@type":"WebSite","@id":"https:\/\/nenadnoveljic.com\/blog\/#website","url":"https:\/\/nenadnoveljic.com\/blog\/","name":"All-round Database Topics","description":"Nenad Noveljic","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/nenadnoveljic.com\/blog\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/nenadnoveljic.com\/blog\/#\/schema\/person\/51458d9dd86dbbdd19f5add451d44efa","name":"Nenad Noveljic","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/a97b796613ea48ec8a7b79c8ffe1c685dcffc920c68121f6238d5caab5070670?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/a97b796613ea48ec8a7b79c8ffe1c685dcffc920c68121f6238d5caab5070670?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/a97b796613ea48ec8a7b79c8ffe1c685dcffc920c68121f6238d5caab5070670?s=96&d=mm&r=g","caption":"Nenad Noveljic"},"sameAs":["nenad-noveljic-9b746a6","https:\/\/x.com\/NenadNoveljic"],"url":"https:\/\/nenadnoveljic.com\/blog\/author\/nenad\/"}]}},"_links":{"self":[{"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/posts\/3055","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/comments?post=3055"}],"version-history":[{"count":1,"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/posts\/3055\/revisions"}],"predecessor-version":[{"id":3159,"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/posts\/3055\/revisions\/3159"}],"wp:attachment":[{"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/media?parent=3055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/categories?post=3055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nenadnoveljic.com\/blog\/wp-json\/wp\/v2\/tags?post=3055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}