{"id":763,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=763"},"modified":"2026-03-09T19:02:50","modified_gmt":"2026-03-09T19:02:50","slug":"cumulative-distribution-function-interpretation-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/cumulative-distribution-function-interpretation-example\/","title":{"rendered":"Cumulative Distribution Function (CDF)"},"content":{"rendered":"\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-107.jpg\",\n  \"caption\": \"Cumulative Distribution\",\n  \"width\": 1463,\n  \"height\": 820,\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  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Variable X can take the values 1, 2, 3, and 4. The probability of each outcome has been given below. Construct a table showing the cumulative distribution and use it to determine P(X \u2264 2).\",\n    \"text\": \"Variable X can take the values 1, 2, 3, and 4 with probabilities 0.2, 0.3, 0.35, and 0.15 respectively. Construct the cumulative distribution table and determine P(X \u2264 2).\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A (0.5). The cumulative probabilities are obtained by summing probabilities up to each outcome: 0.2, 0.5, 0.85, and 1. Therefore, P(X \u2264 2) = 0.2 + 0.3 = 0.5.\"\n    }\n  }\n}\n<\/script>\n\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 used in finance. Topics include discrete vs. continuous random variables, probability functions, binomial and normal distributions, shortfall risk, the safety-first criterion, and the use of Monte Carlo simulations for risk modeling. It also explores log-normal distributions, continuous compounding, and practical applications for 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=etESy1WNTWZwR3I_\" 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>A cumulative distribution function, F(x), gives the probability that the random variable X is less than or equal to x:<\/p>\n<p>$$ P(X \u2264 x) $$<\/p>\n<p>By analogy, this concept is very similar to the cumulative relative frequency.<\/p>\n<p><!--more--><\/p>\n<p>A cumulative distribution is the sum of the probabilities of all values qualifying as &#8220;less than or equal to&#8221; the specified value. Perhaps an example will make this concept clearer.<\/p>\n<h3>Example: Cumulative Distribution<\/h3>\n<p>Imagine flipping a coin three times. You would end up with the following probability distribution of the number of heads obtained:<\/p>\n<p>$$ \\begin{array}{c|c|c|c} \\text{Heads (outcomes)} &amp; {0} &amp; {1} &amp; {2} &amp; {3} \\\\ \\hline \\text{Probability} &amp; {1\/8} &amp; {3\/8} &amp; {3\/8} &amp; {1\/8} \\\\ \\end{array} $$<\/p>\n<p>To come up with a cumulative distribution function, we have to calculate the cumulative probabilities.<\/p>\n<p>The cumulative probability that X is less than or equal to zero is 1\/8.<\/p>\n<p>To find the cumulative probability that X is less than or equal to 1, we add P(X = 0) and (P = 1):<\/p>\n<p>$$ P(X \\le 1) =\\cfrac {1}{8} + \\cfrac {3}{8} = \\cfrac {1}{2} $$<\/p>\n<p>Similarly,<\/p>\n<p>$$ P(X \\le 2) = \\cfrac {1}{8} + \\cfrac {3}{8} + \\cfrac {3}{8} = \\cfrac {7}{8} $$<\/p>\n<p>Lastly,<\/p>\n<p>$$ P(X \\le 3) = \\cfrac {1}{8} + \\cfrac {3}{8} + \\cfrac {3}{8} +\\cfrac {1}{8} = 1 $$<\/p>\n<p>$$ \\begin{array}{c|c|c|c} \\text{Heads (outcomes)} &amp; {0} &amp; {1} &amp; {2} &amp; {3} \\\\ \\hline \\text{Probability} &amp; {1\/8} &amp; {3\/8} &amp; {3\/8} &amp; {1\/8} \\\\ \\hline \\text{Cumulative prob.} &amp; {1\/8} &amp; {4\/8} &amp; {7\/8} &amp; {8\/8} \\\\ \\end{array} $$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-16971\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-107.jpg\" alt=\"cumulative-distribution\" width=\"1463\" height=\"820\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-107.jpg 1463w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-107-300x168.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-107-768x430.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-107-1024x574.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-107-400x224.jpg 400w\" sizes=\"auto, (max-width: 1463px) 100vw, 1463px\" \/><\/p>\n<p>The CDF has two main properties:<\/p>\n<ol>\n<li>all values in the CDF are between 0 and 1; and<\/li>\n<li>the CDF either increases or remains constant as the value of the specified outcome increases.<\/li>\n<\/ol>\n<h2><strong>Interpreting the Cumulative Distribution Function<\/strong><\/h2>\n<p>A cumulative distribution function can help us to come up with cumulative probabilities pretty easily. For example, we can use it to determine the probability of getting at least two heads, at most two heads, or even more than two heads. For example, the probability of at most two heads from the cumulative distribution above is 0.875.<\/p>\n<blockquote>\n<h2>Question<\/h2>\n<p>Variable X can take the values 1, 2, 3, and 4. The probability of each outcome has been given below.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\text{Outcome} &amp; {1} &amp; {2} &amp; {3} &amp; {4} \\\\ \\hline \\text{Probability} &amp; {0.2} &amp; {0.3} &amp; {0.35} &amp; {0.15} \\\\ \\end{array} $$<\/p>\n<p>Construct a table showing the cumulative distribution and use it to determine P(X \u2264 2).<\/p>\n<p>A. 0.5<\/p>\n<p>B. 0.3<\/p>\n<p>C. 0.85<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>A<\/strong>.<\/p>\n<p>You simply sum up the probabilities up to and including a given outcome and come up with a table similar to the one below:<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\text{Heads (outcomes)} &amp; {1} &amp; {2} &amp; {3} &amp; {4} \\\\ \\hline \\text{Probability} &amp; {0.2} &amp; {0.3} &amp; {0.35} &amp; {0.15} \\\\ \\hline \\text{Cumulative prob.} &amp; {0.2} &amp; {0.5} &amp; {0.85} &amp; {1} \\\\ \\end{array} $$<\/p>\n<p>From the table, it is clear that \\(P(X \\le 2) = 0.5\\).<\/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>A cumulative distribution function, F(x), gives the probability that the random variable X is less than or equal to x: $$ P(X \u2264 x) $$ By analogy, this concept is very similar to the cumulative relative frequency.<\/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-763","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 - 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