{"id":1569,"date":"2021-08-25T16:41:40","date_gmt":"2021-08-25T16:41:40","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1569"},"modified":"2023-12-05T09:16:23","modified_gmt":"2023-12-05T09:16:23","slug":"summary-the-exponential-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-the-exponential-distribution\/","title":{"raw":"Summary: The Exponential Distribution","rendered":"Summary: The Exponential Distribution"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\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.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The Poisson distribution is an example of an exponential distribution.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>decay parameter:\u00a0<\/strong>the rate at which probabilities decay to zero for increasing values of [latex]x[\/latex]. It is the value m in the probability density function [latex]f(x)=me^{(-mx)}[\/latex]\u00a0of an exponential random variable. It is also equal to [latex]m=\\frac{1}{\\mu}[\/latex], where [latex]\\mu[\/latex] is the mean of the random variable.\r\n\r\n<strong>exponential distribution:\u00a0<\/strong>a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is [latex]X \\sim Exp(m)[\/latex]. The mean is [latex]\\mu = \\frac{1}{m}[\/latex] and the standard deviation is [latex]\\sigma = \\frac{1}{m}[\/latex]. The probability density function is [latex]f(x)=me^{(-mx)}, x \\geq 0[\/latex] and the cumulative distribution function is [latex]P(X \\leq x) = 1-e^{(-mx)}[\/latex].\r\n\r\n<strong>memoryless property:\u00a0<\/strong>For an exponential random variable [latex]X[\/latex], the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability that [latex]X[\/latex] exceeds [latex]x + k[\/latex], given that it has exceeded [latex]x[\/latex], is the same as the probability that [latex]X[\/latex] would exceed [latex]k[\/latex] if we had no knowledge about it. In symbols we say that [latex]P(X &gt; x + k|X &gt; x) = P(X &gt; k)[\/latex].\r\n\r\n<strong>Poisson distribution:\u00a0<\/strong>If there is a known average of [latex]\u03bb[\/latex] events occurring per unit time, and these events are independent of each other, then the number of events [latex]X[\/latex] occurring in one unit of time has the Poisson distribution. The probability of [latex]k[\/latex] events occurring in one unit time is equal to [latex]P(X=k)= \\frac{\\lambda^{k} e^{- \\lambda}}{k!}[\/latex].","rendered":"<h2>Key Concepts<\/h2>\n<ul>\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.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The Poisson distribution is an example of an exponential distribution.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>decay parameter:\u00a0<\/strong>the rate at which probabilities decay to zero for increasing values of [latex]x[\/latex]. It is the value m in the probability density function [latex]f(x)=me^{(-mx)}[\/latex]\u00a0of an exponential random variable. It is also equal to [latex]m=\\frac{1}{\\mu}[\/latex], where [latex]\\mu[\/latex] is the mean of the random variable.<\/p>\n<p><strong>exponential distribution:\u00a0<\/strong>a continuous random variable (RV) that appears when we are interested in the intervals of time between some random events, for example, the length of time between emergency arrivals at a hospital; the notation is [latex]X \\sim Exp(m)[\/latex]. The mean is [latex]\\mu = \\frac{1}{m}[\/latex] and the standard deviation is [latex]\\sigma = \\frac{1}{m}[\/latex]. The probability density function is [latex]f(x)=me^{(-mx)}, x \\geq 0[\/latex] and the cumulative distribution function is [latex]P(X \\leq x) = 1-e^{(-mx)}[\/latex].<\/p>\n<p><strong>memoryless property:\u00a0<\/strong>For an exponential random variable [latex]X[\/latex], the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability that [latex]X[\/latex] exceeds [latex]x + k[\/latex], given that it has exceeded [latex]x[\/latex], is the same as the probability that [latex]X[\/latex] would exceed [latex]k[\/latex] if we had no knowledge about it. In symbols we say that [latex]P(X > x + k|X > x) = P(X > k)[\/latex].<\/p>\n<p><strong>Poisson distribution:\u00a0<\/strong>If there is a known average of [latex]\u03bb[\/latex] events occurring per unit time, and these events are independent of each other, then the number of events [latex]X[\/latex] occurring in one unit of time has the Poisson distribution. The probability of [latex]k[\/latex] events occurring in one unit time is equal to [latex]P(X=k)= \\frac{\\lambda^{k} e^{- \\lambda}}{k!}[\/latex].<\/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-1569\">\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":15,"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-1569","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1569","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":11,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1569\/revisions"}],"predecessor-version":[{"id":3624,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1569\/revisions\/3624"}],"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\/1569\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1569"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1569"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1569"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1569"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}