{"id":29710,"date":"2021-09-08T16:36:15","date_gmt":"2021-09-08T16:36:15","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29710"},"modified":"2026-01-09T11:35:40","modified_gmt":"2026-01-09T11:35:40","slug":"properties-of-continous-uniform-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/properties-of-continous-uniform-distribution\/","title":{"rendered":"Properties of Continuous Uniform Distribution"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2025 Level I CFA\u00ae Exam \u2013 Quantitative Methods \u2013 Module 4)\",\n  \"description\": \"This video covers CFA Level 1 Quantitative Methods, focusing on probability distributions, including discrete, continuous, normal, Bernoulli, and binomial distributions. It also explores Monte Carlo simulations, z-tables, and lognormal distributions, emphasizing their use in financial modeling, risk analysis, and investment decisions, with practical examples for exam preparation.\",\n  \"uploadDate\": \"2021-11-09T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/TXO9ODcLWiU\/maxresdefault.jpg\",\n  \"contentUrl\": \"https:\/\/youtu.be\/TXO9ODcLWiU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU\",\n  \"duration\": \"PT1H40S\"\n}\n<\/script>\n\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\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1.jpg\",\n  \"caption\": \"Image showing uniform distribution.\",\n  \"width\": 1397,\n  \"height\": 1025,\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\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\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF.jpg\",\n  \"caption\": \"Image of the 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\": \"QAPage\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/properties-of-continous-uniform-distribution\/#qapage-question-1\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/properties-of-continous-uniform-distribution\/#question-1\",\n    \"name\": \"What is the probability that X lies between 32 and 40 for a uniform distribution on [32, 42]?\",\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?\\nA. 8%.\\nB. 10%.\\nC. 80%.\",\n    \"answerCount\": 1,\n    \"author\": {\n      \"@type\": \"Organization\",\n      \"name\": \"AnalystPrep\"\n    },\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/properties-of-continous-uniform-distribution\/#answer-1\",\n      \"text\": \"C. 80%. For a continuous uniform distribution on [32, 42], the probability density is 1\/(42\u221232)=1\/10. Thus, P(32<X<40)=(40\u221232)\/10=0.8, or 80%.\",\n      \"author\": {\n        \"@type\": \"Organization\",\n        \"name\": \"AnalystPrep\"\n      }\n    }\n  }\n}\n<\/script>\n\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The continuous uniform distribution is such that the random variable \\(X\\) takes values between \\(a\\) (lower limit) and \\(b\\) (upper limit). In the field of statistics, \\(a\\) and \\(b\\) are known as the parameters of continuous uniform distribution. We cannot have an outcome of either less than \\(a\\) or greater than \\(b\\).<\/p>\n<p><!--more--><\/p>\n<p>The probability density function for this type of distribution is:<\/p>\n<p>$$ { f }_{ x }\\left( x \\right) =\\frac { 1 }{ b -a } \\quad \\quad a&lt; x &lt; b $$<\/p>\n<p>\\(X \\sim U (a,b)\\) is the most commonly used shorthand notation read as \u201cthe random variable \\(x\\). It has a continuous uniform distribution with parameters \\(\u03b1\\) and \\(\u03b2\\).\u201d<\/p>\n<p>The total probability is spread <strong>uniformly<\/strong> between the two limits. Intervals of the same length have the same probability.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-38204 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1.jpg\" alt=\"Image showing uniform distribution\" width=\"1397\" height=\"1025\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1.jpg 1397w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1-300x220.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1-1024x751.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1-768x563.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-1-400x293.jpg 400w\" sizes=\"auto, (max-width: 1397px) 100vw, 1397px\" \/><\/p>\n<p>In other words, f<span style=\"color: initial;\">or all \\(a \\le x_1 &lt; x_2 \\le b\\), we have<\/span><\/p>\n<p>$$P(X &lt; a) = P(X &gt; b = 0)$$<\/p>\n<p>And the probability that the random variable will have a value between \\(x_1\\) and \\(x_2\\) is given as follows:<\/p>\n<p>$$P(x_1 \\le X \\le x_2) =\\cfrac {(x_2 - x_1)}{(b \u2013 a)}$$<\/p>\n<p>The mean and variance of continuous uniform distribution is given below:<\/p>\n<p>$$ \\text{Mean} =\\cfrac {(a + b)}{2} $$<\/p>\n<p>$$ \\text{Variance} =\\cfrac {(b \u2013 a)^2}{12} $$<\/p>\n<h4><strong>Example: Probability Density Function<\/strong><\/h4>\n<p>You have been given that \\(Y \\sim U(100,300)\\).<\/p>\n<p>Calculate \\(P(Y &gt; 174)\\) and \\(P(100 &lt; Y &lt; 226\\).<\/p>\n<p><strong>Solution<\/strong><\/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>This 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<h3>Cumulative Distribution Function of a Continuous Uniform Distribution<\/h3>\n<p>Remember that a cumulative distribution function, \\(F(x)\\), gives the probability that the random variable \\(X\\) is less than or equal to \\(x\\), for every \\(x\\) value. It is usually expressed as:<\/p>\n<p>$$F(x)=P(X\\leq x)$$<\/p>\n<p>The cumulative distribution function of the continuous uniform distribution increases linearly from \\(\u03b1\\) to \\(\u03b2\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-38202 size-full\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF.jpg\" alt=\"Image of the cumulative distribution function\" width=\"1463\" height=\"932\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF-300x191.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF-1024x652.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF-768x489.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2021\/09\/cfa-level-1-continuous-uniform-random-variable-CDF-400x255.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/p>\n<p>The CDF is linear over the variable\u2019s range, and it is given by:<\/p>\n<p>$$<br \/>F(x)=P(X\\leq x)=\\frac{x-a}{b-a}<br \/>$$<\/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>8%.<\/li>\n<li>10%.<\/li>\n<li>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>","protected":false},"excerpt":{"rendered":"<p>The continuous uniform distribution is such that the random variable \\(X\\) takes values between \\(a\\) (lower limit) and \\(b\\) (upper limit). In the field of statistics, \\(a\\) and \\(b\\) are known as the parameters of continuous uniform distribution. We cannot&#8230;<\/p>\n","protected":false},"author":15,"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-29710","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 Explained | CFA Level 1<\/title>\n<meta name=\"description\" content=\"The continuous uniform distribution assigns equal probability to all outcomes between its lower and upper limits, with mean (a+b)\/2 and variance (b-a)\u00b2\/12.\" \/>\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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