{"id":2701,"date":"2021-11-12T13:31:22","date_gmt":"2021-11-12T13:31:22","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=2701"},"modified":"2023-12-05T09:17:07","modified_gmt":"2023-12-05T09:17:07","slug":"putting-it-together-continuous-random-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/putting-it-together-continuous-random-variables\/","title":{"raw":"Putting It Together: Continuous Random Variables","rendered":"Putting It Together: Continuous Random Variables"},"content":{"raw":"<h2>Let\u2019s Summarize<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">When we have a quantitative variable with outcomes that occur as a result of some random process (e.g., rolling a die, choosing a person at random), we call it a <em>random variable<\/em>.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Continuous random variables can take any value in an interval and are often measurements. We use a density curve to assign probabilities to intervals of <em>x<\/em>-values. We use the area under the density curve to find probabilities.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The probability of an event happening between two numbers [latex]a[\/latex] and [latex]b[\/latex] is written as [latex]P(a\u2264x\u2264b)[\/latex].<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The total area under a continuous probability distribution function is 1.<\/li>\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. Areas of rectangles are used to calculate probabilities associated with uniform distributions.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Exponential probability distributions often follow a decay model with higher probabilities happening for small values and lower probabilities happening for larger values. The Poisson distribution is an example of an exponential distribution.<\/li>\r\n<\/ul>","rendered":"<h2>Let\u2019s Summarize<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">When we have a quantitative variable with outcomes that occur as a result of some random process (e.g., rolling a die, choosing a person at random), we call it a <em>random variable<\/em>.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Continuous random variables can take any value in an interval and are often measurements. We use a density curve to assign probabilities to intervals of <em>x<\/em>-values. We use the area under the density curve to find probabilities.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The probability of an event happening between two numbers [latex]a[\/latex] and [latex]b[\/latex] is written as [latex]P(a\u2264x\u2264b)[\/latex].<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The total area under a continuous probability distribution function is 1.<\/li>\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. Areas of rectangles are used to calculate probabilities associated with uniform distributions.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Exponential probability distributions often follow a decay model with higher probabilities happening for small values and lower probabilities happening for larger values. The Poisson distribution is an example of an exponential distribution.<\/li>\n<\/ul>\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-2701\">\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>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":16,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2701","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2701","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\/2701\/revisions"}],"predecessor-version":[{"id":3626,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/2701\/revisions\/3626"}],"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\/2701\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=2701"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=2701"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=2701"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=2701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}