{"id":25430,"date":"2021-03-12T00:00:03","date_gmt":"2021-03-12T00:00:03","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25430"},"modified":"2024-06-05T04:19:48","modified_gmt":"2024-06-05T04:19:48","slug":"what-is-a-sigma-algebra-definitions-and-examples","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/what-is-a-sigma-algebra-definitions-and-examples\/","title":{"rendered":"What is a Sigma-Algebra? Definitions and Examples"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>What is a Sigma-Algebra? Definition and Examples<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summary<\/strong><br \/><em>This class discusses the importance of sigma-algebra in probability theory. Sigma-algebra is a structure that contains all measurable events in a sample space, enabling the definition of a probability measure. Practical examples, such as coin tosses and the lifespan of an electronic device, explain how sigma-algebra is constructed from the parts of a sample space. The Borel sigma-algebra, associated with a continuous sample space, is also presented, and its Borel events are explained.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nUpon completion of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the definition and characteristics of a Sigma-Algebra, as a mathematical structure that enables the definition of a probability measure.<\/li>\n<li><strong>Identify<\/strong> the elements that make up a Sigma-Algebra and its relationship with measurable events in a sample space.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong>TABLE OF CONTENTS<\/strong><br \/>\n<a href=\"#1\">DEFINITION OF A SIGMA-ALGEBRA<\/a><br \/>\n<a href=\"#2\">SIGMA-ALGEBRA IN COIN TOSSING<\/a><br \/>\n<a href=\"#3\">SIGMA-ALGEBRAS IN CONTINUOUS CASES<\/a><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/cBnDYSiuyfE\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=121s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">Measurable events appear in the probability space through the sigma-algebra.<\/span><\/strong><\/a> Through this idea, an initially intuitive notion is transformed into a formally mathematical structure that allows defining a probability measure.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Definition of a Sigma-Algebra<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=209s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">A Sigma-Algebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma<\/span> (or \u03c3-algebra) is a structure that contains all measurable events of a sample space.<\/span><\/strong><\/a> The pair <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{\\Omega} = (\\Omega, \\mathcal{A}_{\\Omega})<\/span> is said to be a \u03c3-algebra of a <strong><a href=\"https:\/\/toposuranos.com\/el-espacio-de-probabilidades-el-espacio-muestral\/\" rel=\"noopener\" target=\"_blank\">sample space<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> if it satisfies the following:<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset,\\Omega \\in \\mathcal{A}_\\Omega<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\left(E \\in \\mathcal{A}_\\Omega \\right) \\rightarrow (E^c = \\Omega\\setminus E \\in \\mathcal{A}_\\Omega)<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\left(E_1, E_2 \\in \\mathcal{A}_\\Omega \\right) \\rightarrow (E_1 \\cup E_2 \\in \\mathcal{A}_\\Omega)<\/span><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">All objects <span class=\"katex-eq\" data-katex-display=\"false\">E\\in\\mathcal{A}_\\Omega<\/span> are called <strong>Events of <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The sigma-algebra in coin tosses<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXAMPLE 1<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; color: #000000; background-color: #ffffff;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=321s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">For a coin toss,<\/span><\/strong><\/a> the \u03c3-algebra is given by <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{1m}=(\\Omega_{1m}, \\mathcal{A}_{1m})<\/span>, where<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{H,T\\}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}= \\{\\emptyset,\\{H\\},\\{T\\}, \\Omega_{1m}\\}<\/span><\/li>\n<\/ul>\n<p>Each element of <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{1m}<\/span> is an event identified as follows:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abNeither heads nor tails appear\u00bb (it is the impossible event).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{H\\}<\/span> \u00abIt is the event in which heads appear\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{T\\}<\/span> \u00abIt is the event in which tails appear\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{1m}= \\{H,T\\}<\/span> \u00abEither heads or tails appears\u00bb (it is the certain event).<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXAMPLE 2<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; color: #000000; background-color: #ffffff;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=558s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">If we toss two coins instead of one, <\/span><\/strong><\/a>a possible \u03c3-algebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_{2m}=(\\Omega_{2m}, \\mathcal{A}_{2m})<\/span> can be obtained from the parts of <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span>. Thus we have the following:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}= \\{(C,C);(C,S); (S,C); (S,S)\\}<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_{2m}=\\mathcal{P}(\\Omega_{2m}) = \\cdots<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\cdots = \\{\\emptyset; \\{(C,C)\\}; \\left\\{(C,S)\\}; \\{(S,C)\\}; \\{(S,S)\\}; \\cdots \\right. <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cdots; \\{(C,C);(C,S)\\};\\{(C,C);(S,C)\\};\\{(C,C);(S,S)\\};\\cdots<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cdots; \\{(C,S);(S,C)\\};\\{(C,S);(S,S)\\};\\{(S,C);(S,S)\\};\\cdots<\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\cdots; \\{(C,C);(C,S);(S,C)\\};\\{(C,C);(C,S);(S,S)\\}\\cdots<\/span>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\left. \\cdots; {(C,C);(S,C);(S,S)}; {(C,S);(S,C);(S,S)}; \\Omega_{2m}\\right\\}<\/span><\/li>\n<\/ul>\n<p>Each element of <span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}{2m}<\/span> is an event of <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega{2m}<\/span>. Here are some of them:<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\emptyset<\/span> \u00abNo results occur\u00bb (the impossible event).<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C)\\}<\/span> \u00abTwo consecutive heads\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,S)\\}<\/span> \u00abHead followed by tail\u00bb.<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(C,S)\\}<\/span> \u00abFirst coin shows heads, second coin can show any result\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C)\\}<\/span> \u00abFirst coin can show any result, second coin shows heads\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,S)\\}<\/span> \u00abBoth tosses have the same result\u00bb.<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(C,S);(S,S)\\}<\/span> \u00abIf the first toss shows tails, then the second toss also shows tails; otherwise, the second toss can show any result\u00bb.<\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\{(C,C);(S,C);(S,S)\\}<\/span> \u00abIf the first toss shows heads, then the second toss also shows heads; otherwise, the second toss can show any result\u00bb.<br \/>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2m}<\/span> \u00abAny possible result can occur\u00bb (the certain event).<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Sigma-algebras in Continuous Cases<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: justify; color: #008000; background-color: #dddddd;\"><strong>EXAMPLE 3<\/strong><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: justify; color: #000000; background-color: #ffffff;\"><a href=\"https:\/\/www.youtube.com\/watch?v=cBnDYSiuyfE&amp;t=1265s\" rel=\"noopener\" target=\"_blank\"><strong><span style=\"color: #d00000;\">For the lifetime (measured in hours) of an electronic device that could fail at any moment,<\/span><\/strong><\/a> the \u03c3-algebra <span class=\"katex-eq\" data-katex-display=\"false\">\\Sigma_e = (\\Omega_e, \\mathcal{A}_e)<\/span> is given by<\/p>\n<ul>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_e = [0, \\infty[ <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{A}_e = \\{I \\; | \\; I \\subseteq \\Omega_e \\}<\/span><\/li>\n<\/ul>\n<p>Thus, the intervals <span class=\"katex-eq\" data-katex-display=\"false\">I_t = ]0,t[\\in\\mathcal{A}_e <\/span> can be interpreted as \u00abthe electronic device works correctly for an interval of <span class=\"katex-eq\" data-katex-display=\"false\">t<\/span> consecutive hours until it fails.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">The probability \u03c3-algebra associated with a continuous sample space is also known as the <strong>Borel \u03c3-algebra<\/strong> and its events are known as Borel sets.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is a Sigma-Algebra? Definition and Examples SummaryThis class discusses the importance of sigma-algebra in probability theory. Sigma-algebra is a structure that contains all measurable events in a sample space, enabling the definition of a probability measure. Practical examples, such as coin tosses and the lifespan of an electronic device, explain how sigma-algebra is constructed [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25427,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":35,"footnotes":""},"categories":[567,670],"tags":[],"class_list":["post-25430","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-probabilities-and-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>What is a Sigma-Algebra? 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