{"id":1214,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1214"},"modified":"2026-03-08T18:29:02","modified_gmt":"2026-03-08T18:29:02","slug":"continuous-uniform-distribution-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/continuous-uniform-distribution-example\/","title":{"rendered":"Continuous Uniform Distribution"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"A random variable X is uniformly distributed between 32 and 42. What is the probability that X will be between 32 and 40?\",\n    \"text\": \"A random variable X is uniformly distributed between 32 and 42. What is the probability that X will be between 32 and 40?\\n\\nA. 8%\\n\\nB. 10%\\n\\nC. 80%\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. For a uniform distribution between 32 and 42, the probability density function is f(x) = 1 \/ (42 \u2212 32) = 1\/10. The probability that X lies between 32 and 40 equals the length of that interval divided by the total range: (40 \u2212 32) \/ 10 = 8\/10 = 0.8, or 80%.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121.jpg\",\n  \"caption\": \"Cumulative Distribution Function\",\n  \"width\": 1463,\n  \"height\": 932,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2021 Level I CFA\u00ae Exam \u2013 Reading 9)\",\n  \"description\": \"This video lesson covers common probability distributions, including discrete and continuous variables, Bernoulli and binomial trials, and normal distributions. It explains how to calculate probabilities, interpret distributions, construct binomial trees, apply the safety-first criterion, and utilize Monte Carlo simulations and continuous compounding in financial analysis.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/i.ytimg.com\/vi\/KbEfz3KiJDo\/hqdefault.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=KbEfz3KiJDo\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo\",\n  \"duration\": \"PT45M31S\"\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?si=zpLEVcRkIuWRms7_\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n<p>The continuous uniform distribution is such that the random variable X takes values between \u03b1 (lower limit) and \u03b2 (upper limit). In the field of statistics, \u03b1 and \u03b2 are known as the parameters of the continuous uniform distribution. We cannot have an outcome of either less than \u03b1 or greater than \u03b2.<\/p>\n<p><!--more--><\/p>\n<p>The probability density function for this type of distribution is:<\/p>\n<p>$$ { f }_{ x }\\left( x \\right) =\\frac { 1 }{ \\beta -\\alpha } \\quad \\quad \\alpha &lt; x &lt; \\beta $$<\/p>\n<p>\\(X \\sim U (\\alpha, \\beta)\\) is the most commonly used shorthand notation read as \u201cthe random variable x has a continuous uniform distribution with parameters \u03b1 and \u03b2.\u201d<\/p>\n<p>The total probability (1) is spread <strong>uniformly<\/strong> between the two limits. <strong>Intervals of the same length have the same probability<\/strong>.<\/p>\n<p><strong>Properties of the Continuous Uniform Distribution<\/strong><\/p>\n<ol>\n<li>For all \\(\\alpha \\le x_1 &lt; x_2 \\le \\beta\\)<\/li>\n<li>\\(P(X &lt; \\alpha) = P(X &gt; \\beta = 0)\\)<\/li>\n<li>\\(P(x_1 \\le X \\le x_2) =\\cfrac {(x_2 &#8211; x_1)}{(\\beta \u2013 \\alpha)}\\)<\/li>\n<\/ol>\n<h2><strong>Moments<\/strong><\/h2>\n<p>$$ \\text{Mean} =\\cfrac {(\\alpha + \\beta)}{2} $$<\/p>\n<p>$$ \\text{Variance} =\\cfrac {(\\beta \u2013 \\alpha)^2}{12} $$<\/p>\n<h3>Example: Probability Density Function<\/h3>\n<p>You have been given that \\(Y \\sim U(100,300)\\). Calculate \\(P(Y &gt; 174)\\) and \\(P(100 &lt; Y &lt; 226\\)<\/p>\n<p>The probability density function is given by:<\/p>\n<p>$$ f_x(x) =\\cfrac {1}{(300 \u2013 100)} =\\cfrac {1}{200} $$<\/p>\n<p>Therefore, each \u201cunit interval\u201d has a probability of \\(\\frac {1}{200}\\).<\/p>\n<p>Which means that \\(P(Y &gt; 174) =\\cfrac {(300 \u2013 174)}{200} = \\cfrac {126}{200} = 0.63\\)<\/p>\n<p>Similarly, \\(P(100 &lt; Y &lt; 226) = 0.63\\) because the interval has the same length as above (126) hence the same probability.<\/p>\n<p>The cumulative distribution function of the continuous uniform distribution looks like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-16992\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121.jpg\" alt=\"\" width=\"1463\" height=\"932\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121-300x191.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121-768x489.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121-1024x652.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-121-400x255.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/p>\n<p>The CDF is linear over the variable\u2019s range.<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>A random variable X is uniformly distributed between 32 and 42. What is the probability that X will be between 32 and 40?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li data-tadv-p=\"keep\">8%<\/li>\n<li data-tadv-p=\"keep\">10%<\/li>\n<li data-tadv-p=\"keep\">80%<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>First, you should determine the pdf:<\/p>\n<p>$$ \\begin{align*} f_x(x) &amp; =\\cfrac {1}{(42 \u2013 32)} \\\\ &amp;=\\cfrac {1}{10} \\\\ \\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ P(32 &lt; Y &lt; 40) =\\cfrac {(40 \u2013 32)}{10} = 0.8 \\text{ or } 80\\%. $$<\/p>\n<\/blockquote>\n<div class=\"notes_inv\"><hr \/>\n<p><a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/learning-sessions-curriculum\/\"><em>Quantitative Methods \u2013 Learning Sessions<\/em><\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>The continuous uniform distribution is such that the random variable X takes values between \u03b1 (lower limit) and \u03b2 (upper limit). In the field of statistics, \u03b1 and \u03b2 are known as the parameters of the continuous uniform distribution. We&#8230;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-1214","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Continuous Uniform Distribution Example | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Understand the continuous uniform distribution, where a random variable takes values between defined upper and lower limits.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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