{"id":1292,"date":"2021-08-23T19:45:42","date_gmt":"2021-08-23T19:45:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1292"},"modified":"2023-12-05T09:10:54","modified_gmt":"2023-12-05T09:10:54","slug":"summary-geometric-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-geometric-distribution\/","title":{"raw":"Summary: Geometric Distribution","rendered":"Summary: Geometric Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The probability of success and failure is the same in each trial of a geometric experiment.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">In a geometric experiment, [latex]X =[\/latex] the number of independent trials until the first success.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The number of trials in a geometric experiment is not fixed.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>geometric distribution:\u00a0<\/strong>a discrete random variable [latex](RV)[\/latex] that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable [latex]X[\/latex] is defined as the number of trials until the first success. Notation: [latex]X \\sim G(p)[\/latex]. The mean is [latex]\\mu = \\frac{1}{p}[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{\\frac{1}{p} (\\frac{1}{p} - 1)}[\/latex]. The probability of exactly [latex]x[\/latex] failures before the first success is given by the formula: [latex]P(X=x)=p(1-p)^{x-1}[\/latex].\r\n\r\n<strong>geometric experiment:\u00a0<\/strong>a statistical experiment with the following properties:\r\n<ol>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">There are one or more Bernoulli trials with all failures except the last one, which is a success.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">In theory, the number of trials could go on forever. There must be at least one trial.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The probability, [latex]p[\/latex], of a success and the probability, [latex]q[\/latex], of a failure do not change from trial to trial.<\/li>\r\n<\/ol>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The probability of success and failure is the same in each trial of a geometric experiment.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">In a geometric experiment, [latex]X =[\/latex] the number of independent trials until the first success.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The number of trials in a geometric experiment is not fixed.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>geometric distribution:\u00a0<\/strong>a discrete random variable [latex](RV)[\/latex] that arises from the Bernoulli trials; the trials are repeated until the first success. The geometric variable [latex]X[\/latex] is defined as the number of trials until the first success. Notation: [latex]X \\sim G(p)[\/latex]. The mean is [latex]\\mu = \\frac{1}{p}[\/latex] and the standard deviation is [latex]\\sigma = \\sqrt{\\frac{1}{p} (\\frac{1}{p} - 1)}[\/latex]. The probability of exactly [latex]x[\/latex] failures before the first success is given by the formula: [latex]P(X=x)=p(1-p)^{x-1}[\/latex].<\/p>\n<p><strong>geometric experiment:\u00a0<\/strong>a statistical experiment with the following properties:<\/p>\n<ol>\n<li style=\"font-weight: 400;\" aria-level=\"1\">There are one or more Bernoulli trials with all failures except the last one, which is a success.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">In theory, the number of trials could go on forever. There must be at least one trial.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The probability, [latex]p[\/latex], of a success and the probability, [latex]q[\/latex], of a failure do not change from trial to trial.<\/li>\n<\/ol>\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-1292\">\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\/4-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-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":20,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-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-1292","chapter","type-chapter","status-publish","hentry"],"part":240,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1292","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":4,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1292\/revisions"}],"predecessor-version":[{"id":3572,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1292\/revisions\/3572"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/240"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1292\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1292"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1292"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1292"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}