{"id":1559,"date":"2021-08-25T16:32:24","date_gmt":"2021-08-25T16:32:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1559"},"modified":"2023-12-05T09:15:44","modified_gmt":"2023-12-05T09:15:44","slug":"summary-the-uniform-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-the-uniform-distribution\/","title":{"raw":"Summary: The Uniform Distribution","rendered":"Summary: The Uniform Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">A uniform distribution is a type of continuous random variable, where all outcomes are equally likely on a given range of values.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Areas of rectangles are used to calculate probabilities associated with uniform distributions.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>uniform distribution<\/strong>\u00a0a continuous random variable (RV) that has equally likely outcomes over the domain, [latex]a&lt;x&lt;b[\/latex]. Notation: [latex]X \\sim U(a,b)[\/latex]. The mean is [latex]\\mu = \\frac{a+b}{2}[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{\\frac{(b-a)^{2}}{12}}[\/latex]. The probability density function is [latex]f(x)=\\frac{1}{b-a}[\/latex] for [latex]a&lt;x&lt;b[\/latex] or [latex]a \\leq x \\leq b[\/latex]. The cumulative distribution is [latex]P(X \\leq x) = \\frac{x-a}{b-a}[\/latex].","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">A uniform distribution is a type of continuous random variable, where all outcomes are equally likely on a given range of values.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Areas of rectangles are used to calculate probabilities associated with uniform distributions.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>uniform distribution<\/strong>\u00a0a continuous random variable (RV) that has equally likely outcomes over the domain, [latex]a<x<b[\/latex]. Notation: [latex]X \\sim U(a,b)[\/latex]. The mean is [latex]\\mu = \\frac{a+b}{2}[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{\\frac{(b-a)^{2}}{12}}[\/latex]. The probability density function is [latex]f(x)=\\frac{1}{b-a}[\/latex] for [latex]a<x<b[\/latex] or [latex]a \\leq x \\leq b[\/latex]. The cumulative distribution is [latex]P(X \\leq x) = \\frac{x-a}{b-a}[\/latex].\n<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1559\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/5-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/5-key-terms<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/5-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1559","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1559","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1559\/revisions"}],"predecessor-version":[{"id":3618,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1559\/revisions\/3618"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/249"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1559\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1559"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1559"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1559"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1559"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}