{"id":1820,"date":"2024-05-28T12:10:31","date_gmt":"2024-05-28T16:10:31","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=part&#038;p=1820"},"modified":"2024-07-24T10:19:50","modified_gmt":"2024-07-24T14:19:50","slug":"exponential-distributions","status":"publish","type":"part","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/part\/exponential-distributions\/","title":{"raw":"Exponential Distributions","rendered":"Exponential Distributions"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nUnderstand the shape, statistical properties and probability formulas for exponential distributions.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Properties of Exponential Distributions<\/h2>\r\nAn <a href=\"https:\/\/www.wolframalpha.com\/input?i=exponential+distributions\">exponential distribution<\/a> is:\r\n<ul>\r\n \t<li>A highly used 'continuous' distribution<\/li>\r\n \t<li>Often used to model the time elapsed between events (we call this [latex]x[\/latex]).<\/li>\r\n \t<li>It is 'memoryless' (see more in the 'MEMORYLESS' section below)<\/li>\r\n \t<li>We call the average lambda ([latex]\\lambda[\/latex]). It measures the average number of events per time unit.<\/li>\r\n \t<li>It is right skewed (mean &gt; median), and [latex]\\lambda[\/latex] and [latex]x[\/latex] can never be negative.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_1833\" align=\"aligncenter\" width=\"750\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1.jpg\"><img class=\"wp-image-1833 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1.jpg\" alt=\"Graphs of three different exponential distributions for different values of lambda.\" width=\"750\" height=\"372\" \/><\/a> The exponential distribution for different values of lambda.[\/caption]\r\n<h1>Calculating Probabilities<\/h1>\r\nWe can calculate the probabilities using formulas or Excel. The formulas are:\r\n<ul>\r\n \t<li>[latex]P(\\text{at most or less than}) = P(X \\le x) = 1 - e^{-\\lambda x}[\/latex]<\/li>\r\n \t<li>[latex]P(\\text{at least or more than}) = P(X \\ge x) = e^{-\\lambda x}[\/latex]<\/li>\r\n<\/ul>\r\nThe Excel calculations are:\r\n<ul>\r\n \t<li>[latex]P(\\text{at most or less than}) =\\text{EXPON.DIST}(x, \\lambda, \\text{TRUE})[\/latex]<\/li>\r\n \t<li>[latex]P(\\text{at least or more than}) =1-\\text{EXPON.DIST}(x, \\lambda, \\text{TRUE})[\/latex]<\/li>\r\n<\/ul>\r\n<h1>Memoryless Property<\/h1>\r\n<ul>\r\n \t<li>Memoryless means that it does not matter how much time has elapsed previously.<\/li>\r\n \t<li>When calculating the odds of a certain amount of time elapsing going forward,<\/li>\r\n \t<li>It's although the clock resets to zero.<\/li>\r\n \t<li>See <a href=\"https:\/\/en.wikipedia.org\/wiki\/Memorylessness\">Wikipedia<\/a>'s explanation of memoryless:\r\n<blockquote>It describes situations where the time you've already waited for an event doesn't affect how much longer you'll have to wait<\/blockquote>\r\n<\/li>\r\n<\/ul>\r\n<h1>Statistical Properties<\/h1>\r\nThe following metrics apply to exponential distributions:\r\n<ul>\r\n \t<li>[latex]x = \\text{time between events}[\/latex]<\/li>\r\n \t<li>[latex]\\lambda = \\text{lambda}=\\text{number of events per time unit}[\/latex]<\/li>\r\n \t<li>The mean is: [latex] \\mu = \\frac{1}{\\lambda}[\/latex]<\/li>\r\n \t<li>The standard deviation is also: [latex] \\frac{1}{\\lambda}[\/latex]<\/li>\r\n \t<li>The variance is: [latex] \\sigma^2 = \\frac{1}{\\lambda ^2}[\/latex]<\/li>\r\n \t<li>The distribution is right skewed and the skewness = 2.<\/li>\r\n<\/ul>\r\n<h1>Video Explaining Exponential Distributions<\/h1>\r\nhttps:\/\/youtu.be\/bKkLYSi5XNE\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Exponential Distributions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"102\"]\r\n\r\n[h5p id=\"103\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Understand the shape, statistical properties and probability formulas for exponential distributions.<\/p>\n<\/div>\n<\/div>\n<h2>Properties of Exponential Distributions<\/h2>\n<p>An <a href=\"https:\/\/www.wolframalpha.com\/input?i=exponential+distributions\">exponential distribution<\/a> is:<\/p>\n<ul>\n<li>A highly used &#8216;continuous&#8217; distribution<\/li>\n<li>Often used to model the time elapsed between events (we call this [latex]x[\/latex]).<\/li>\n<li>It is &#8216;memoryless&#8217; (see more in the &#8216;MEMORYLESS&#8217; section below)<\/li>\n<li>We call the average lambda ([latex]\\lambda[\/latex]). It measures the average number of events per time unit.<\/li>\n<li>It is right skewed (mean &gt; median), and [latex]\\lambda[\/latex] and [latex]x[\/latex] can never be negative.<\/li>\n<\/ul>\n<figure id=\"attachment_1833\" aria-describedby=\"caption-attachment-1833\" style=\"width: 750px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1833 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1.jpg\" alt=\"Graphs of three different exponential distributions for different values of lambda.\" width=\"750\" height=\"372\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1.jpg 1015w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1-300x149.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1-768x381.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1-65x32.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1-225x112.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/ExpDists-1-350x174.jpg 350w\" sizes=\"auto, (max-width: 750px) 100vw, 750px\" \/><\/a><figcaption id=\"caption-attachment-1833\" class=\"wp-caption-text\">The exponential distribution for different values of lambda.<\/figcaption><\/figure>\n<h1>Calculating Probabilities<\/h1>\n<p>We can calculate the probabilities using formulas or Excel. The formulas are:<\/p>\n<ul>\n<li>[latex]P(\\text{at most or less than}) = P(X \\le x) = 1 - e^{-\\lambda x}[\/latex]<\/li>\n<li>[latex]P(\\text{at least or more than}) = P(X \\ge x) = e^{-\\lambda x}[\/latex]<\/li>\n<\/ul>\n<p>The Excel calculations are:<\/p>\n<ul>\n<li>[latex]P(\\text{at most or less than}) =\\text{EXPON.DIST}(x, \\lambda, \\text{TRUE})[\/latex]<\/li>\n<li>[latex]P(\\text{at least or more than}) =1-\\text{EXPON.DIST}(x, \\lambda, \\text{TRUE})[\/latex]<\/li>\n<\/ul>\n<h1>Memoryless Property<\/h1>\n<ul>\n<li>Memoryless means that it does not matter how much time has elapsed previously.<\/li>\n<li>When calculating the odds of a certain amount of time elapsing going forward,<\/li>\n<li>It&#8217;s although the clock resets to zero.<\/li>\n<li>See <a href=\"https:\/\/en.wikipedia.org\/wiki\/Memorylessness\">Wikipedia<\/a>&#8216;s explanation of memoryless:<br \/>\n<blockquote><p>It describes situations where the time you&#8217;ve already waited for an event doesn&#8217;t affect how much longer you&#8217;ll have to wait<\/p><\/blockquote>\n<\/li>\n<\/ul>\n<h1>Statistical Properties<\/h1>\n<p>The following metrics apply to exponential distributions:<\/p>\n<ul>\n<li>[latex]x = \\text{time between events}[\/latex]<\/li>\n<li>[latex]\\lambda = \\text{lambda}=\\text{number of events per time unit}[\/latex]<\/li>\n<li>The mean is: [latex]\\mu = \\frac{1}{\\lambda}[\/latex]<\/li>\n<li>The standard deviation is also: [latex]\\frac{1}{\\lambda}[\/latex]<\/li>\n<li>The variance is: [latex]\\sigma^2 = \\frac{1}{\\lambda ^2}[\/latex]<\/li>\n<li>The distribution is right skewed and the skewness = 2.<\/li>\n<\/ul>\n<h1>Video Explaining Exponential Distributions<\/h1>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"The Exponential Distribution Made EASY!\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bKkLYSi5XNE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Exponential Distributions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-102\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-102\" class=\"h5p-iframe\" data-content-id=\"102\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Exponential Distributions Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-103\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-103\" class=\"h5p-iframe\" data-content-id=\"103\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Exponential Distributions Key Takeaways Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"parent":0,"menu_order":8,"template":"","meta":{"pb_part_invisible":false,"pb_part_invisible_string":""},"contributor":[],"license":[],"class_list":["post-1820","part","type-part","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1820","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/part"}],"version-history":[{"count":23,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1820\/revisions"}],"predecessor-version":[{"id":1964,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1820\/revisions\/1964"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1820"}],"wp:term":[{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1820"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1820"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}