{"id":1701,"date":"2024-05-24T09:41:21","date_gmt":"2024-05-24T13:41:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=part&#038;p=1701"},"modified":"2024-07-24T10:19:26","modified_gmt":"2024-07-24T14:19:26","slug":"uniform-distributions","status":"publish","type":"part","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/part\/uniform-distributions\/","title":{"raw":"Continuous Uniform Distributions","rendered":"Continuous Uniform 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 formula for continuous uniform distributions.\r\n\r\n<\/div>\r\n<\/div>\r\n<h2>Continuous Distribution<\/h2>\r\nA <a href=\"https:\/\/www.wolframalpha.com\/input?i=uniform+distribution\">continuous uniform distribution<\/a> is a 'continuous' distribution:\r\n<ul>\r\n \t<li>Any value, [latex]x[\/latex], between the lower [latex]a[\/latex] and upper limit [latex]b[\/latex] is possible<\/li>\r\n \t<li>It differs from 'discrete' distributions where only whole numbers are possible for [latex]x[\/latex].<\/li>\r\n \t<li>See the graphs of uniform distributions with different min and max values below.<\/li>\r\n<\/ul>\r\n<a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions.jpg\"><img class=\"alignnone size-full wp-image-1702\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions.jpg\" alt=\"\" width=\"936\" height=\"361\" \/><\/a>\r\n<h1>Equal Likelihood<\/h1>\r\n<ul>\r\n \t<li>Uniform means <a href=\"https:\/\/www.oed.com\/\">\"remaining the same at all times<\/a>\"<\/li>\r\n \t<li>We see from the above graph that the height, [latex]h[\/latex], remains the same over each uniform distribution.<\/li>\r\n \t<li>This is due to the fact that there is equal likelihood of each value, [latex]x[\/latex], occurring<\/li>\r\n \t<li>This gives each distribution the shape of a rectangle.<\/li>\r\n \t<li>Because of this and the fact that the total area of any probability distribution must equal to 1:\r\n[latex] (b-a) \\times h = 1 [\/latex] or, [latex] h = \\frac{1}{b-a} [\/latex]<\/li>\r\n \t<li>This gives an area (or probability) between two [latex]x[\/latex]-values, [latex]x_1[\/latex] and [latex]x_2[\/latex]:\r\n[latex]P(x_1 \\le x \\le x_2) = (x_2 - x_1) \\times \\frac{1}{b-a} = \\frac{x_2 - x_1}{b-a} [\/latex]<\/li>\r\n<\/ul>\r\n<h1>Statistical Properties<\/h1>\r\nThe following metrics apply to uniform distributions:\r\n<ul>\r\n \t<li>They have a lower limit (lowest possible value): min = [latex]a[\/latex]<\/li>\r\n \t<li>They have an upper limit (highest possible value): max = [latex]b[\/latex]<\/li>\r\n \t<li>The mean is: [latex] \\mu = \\frac{a+b}{2}[\/latex]<\/li>\r\n \t<li>The standard deviation is:\u00a0 [latex] \\sigma = \\frac{b-a}{\\sqrt{12}}[\/latex]<\/li>\r\n \t<li>The variance is: [latex] \\sigma^2 = \\frac{(b-a)^2}{12}[\/latex]<\/li>\r\n \t<li>The distribution is symmetric, so skewness = 0.<\/li>\r\n<\/ul>\r\n<h1>Video Explaining Uniform Distributions<\/h1>\r\nhttps:\/\/youtu.be\/UC-CBUSQXAo?si=PdR1-LE-NwewVUV0\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: Continuous Uniform Distributions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"91\"]\r\n\r\n[h5p id=\"92\"]\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 formula for continuous uniform distributions.<\/p>\n<\/div>\n<\/div>\n<h2>Continuous Distribution<\/h2>\n<p>A <a href=\"https:\/\/www.wolframalpha.com\/input?i=uniform+distribution\">continuous uniform distribution<\/a> is a &#8216;continuous&#8217; distribution:<\/p>\n<ul>\n<li>Any value, [latex]x[\/latex], between the lower [latex]a[\/latex] and upper limit [latex]b[\/latex] is possible<\/li>\n<li>It differs from &#8216;discrete&#8217; distributions where only whole numbers are possible for [latex]x[\/latex].<\/li>\n<li>See the graphs of uniform distributions with different min and max values below.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-1702\" src=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions.jpg\" alt=\"\" width=\"936\" height=\"361\" srcset=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions.jpg 936w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions-300x116.jpg 300w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions-768x296.jpg 768w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions-65x25.jpg 65w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions-225x87.jpg 225w, https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/UniformDistributions-350x135.jpg 350w\" sizes=\"auto, (max-width: 936px) 100vw, 936px\" \/><\/a><\/p>\n<h1>Equal Likelihood<\/h1>\n<ul>\n<li>Uniform means <a href=\"https:\/\/www.oed.com\/\">&#8220;remaining the same at all times<\/a>&#8220;<\/li>\n<li>We see from the above graph that the height, [latex]h[\/latex], remains the same over each uniform distribution.<\/li>\n<li>This is due to the fact that there is equal likelihood of each value, [latex]x[\/latex], occurring<\/li>\n<li>This gives each distribution the shape of a rectangle.<\/li>\n<li>Because of this and the fact that the total area of any probability distribution must equal to 1:<br \/>\n[latex](b-a) \\times h = 1[\/latex] or, [latex]h = \\frac{1}{b-a}[\/latex]<\/li>\n<li>This gives an area (or probability) between two [latex]x[\/latex]-values, [latex]x_1[\/latex] and [latex]x_2[\/latex]:<br \/>\n[latex]P(x_1 \\le x \\le x_2) = (x_2 - x_1) \\times \\frac{1}{b-a} = \\frac{x_2 - x_1}{b-a}[\/latex]<\/li>\n<\/ul>\n<h1>Statistical Properties<\/h1>\n<p>The following metrics apply to uniform distributions:<\/p>\n<ul>\n<li>They have a lower limit (lowest possible value): min = [latex]a[\/latex]<\/li>\n<li>They have an upper limit (highest possible value): max = [latex]b[\/latex]<\/li>\n<li>The mean is: [latex]\\mu = \\frac{a+b}{2}[\/latex]<\/li>\n<li>The standard deviation is:\u00a0 [latex]\\sigma = \\frac{b-a}{\\sqrt{12}}[\/latex]<\/li>\n<li>The variance is: [latex]\\sigma^2 = \\frac{(b-a)^2}{12}[\/latex]<\/li>\n<li>The distribution is symmetric, so skewness = 0.<\/li>\n<\/ul>\n<h1>Video Explaining Uniform Distributions<\/h1>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Uniform Distribution EXPLAINED with Examples\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UC-CBUSQXAo?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: Continuous Uniform Distributions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-91\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-91\" class=\"h5p-iframe\" data-content-id=\"91\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform Distributions Key Takeaways\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-92\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-92\" class=\"h5p-iframe\" data-content-id=\"92\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Continuous Uniform 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":7,"template":"","meta":{"pb_part_invisible":false,"pb_part_invisible_string":""},"contributor":[],"license":[],"class_list":["post-1701","part","type-part","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1701","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":22,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1701\/revisions"}],"predecessor-version":[{"id":1960,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/1701\/revisions\/1960"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1701"}],"wp:term":[{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1701"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1701"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}