{"id":942,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=942"},"modified":"2025-12-30T13:08:34","modified_gmt":"2025-12-30T13:08:34","slug":"calculating-probabilities-given-cumulative-distribution-function","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-probabilities-given-cumulative-distribution-function\/","title":{"rendered":"The Cumulative Distribution Function: Interpretation and Determination of Probabilities"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Given the following cumulative probability distribution, determine P(2).\",\n    \"text\": \"Question\\nGiven the following cumulative probability distribution, determine P(2).\\n\\nOutcome: 0, 1, 2, 3\\nCumulative probabilities: 1\/8, 4\/8, 7\/8, 1\\n\\nA. 7\/8\\nB. 3\/8\\nC. 1\/8\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B. P(2) = F(2) \u2212 F(1) = 7\/8 \u2212 4\/8 = 3\/8.\"\n    }\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?si=rske8g_8sgtCWQO1\" 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 offers a convenient tool for determining probabilities for a given random variable. As you have already learnt in a previous learning outcome statement, a cumulative distribution function, F(x), gives the probability that the random variable X is less than or equal to x for every value x. It is usually expressed as:<\/p>\n<p><!--more--><\/p>\n<p>$$ F(x) = P(X \\le x) $$<\/p>\n<h3><strong>Example 1: Cumulative Distribution Function<\/strong><\/h3>\n<p>The random variable X has the following probability distribution function:<\/p>\n<p>$$ \\begin{matrix} P(x) = \\frac { x }{ 150 } &amp; \\text{ for x} = 10, 20, 30, 40, 50 \\\\ 0 &amp; \\text{otherwise} \\end{matrix} $$<\/p>\n<p>Calculate and interpret F(20) and F(40), giving an interpretation for each.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>As you will recall, we can determine the probability of each outcome for a random variable given the <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/probability-distributions-random-variables-types-examples\/\">probability distribution function (pdf).<\/a><\/p>\n<p>$$ \\begin{align*} P(x) &amp; =\\cfrac {x}{150} \\\\ P(x) &amp; = P(X = x) \\\\ \\end{align*} $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ P(10) =\\cfrac {10}{150} $$<\/p>\n<p>Similarly,<\/p>\n<p>$$ P(20) =\\cfrac {20}{150} $$<\/p>\n<p>$$ P(30) =\\cfrac {30}{150} $$<\/p>\n<p>$$ P(40) =\\cfrac {40}{150} $$<\/p>\n<p>And lastly,<\/p>\n<p>$$ P(50) =\\cfrac {50}{150} $$<\/p>\n<p><em>Note: we can prove that our pdf is correct by testing the first rule of probability distribution functions. To do this, we add all the probabilities.<\/em><\/p>\n<p>Now,<\/p>\n<p>$$ F(x) = P(X \\le x) $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ \\begin{align*} F(2) &amp; = P(X \\le 20) \\\\ &amp; = P(X = 10) + P(X = 30) \\\\ &amp; =\\cfrac {10}{150} + \\cfrac {20}{150} \\\\ &amp;=\\cfrac {30}{150} \\text { or } \\cfrac {1}{5} \\\\ \\end{align*} $$<\/p>\n<p>Interpretation: There is a 20% cumulative probability that outcomes 10 or 20 occur.<\/p>\n<p>Similarly,<\/p>\n<p>$$ \\begin{align*} F(40) &amp; = P(X \\le 40) \\\\ &amp; = P(X = 10) + P(X = 20) + P(X = 30) + P(X = 40) \\\\ &amp; =\\cfrac {10}{150} + \\cfrac {20}{150} + \\cfrac {30}{150} + \\cfrac {40}{150} \\\\ &amp; = \\cfrac {100}{150} \\text{ or } 66.67\\% \\\\ \\end{align*} $$<\/p>\n<p>Interpretation: There is a 66.67% cumulative probability that outcomes 10, 20, 30, or 40 occur.<\/p>\n<h3><strong>Example 2: Cumulative Distribution Function<\/strong><\/h3>\n<p>Variable X can take the values 1, 2, 3, and 4. The cumulative probability distribution has been given below. Use it to calculate:<\/p>\n<p>(a) P(X = 2)<\/p>\n<p>(b) P(X = 4)<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\text{Outcome} &amp; {1} &amp; {2} &amp; {3} &amp; {4} \\\\ \\hline \\text{Cumulative Probability distribution} &amp; {0.2} &amp; {0.5} &amp; {0.85} &amp; {1} \\\\ \\end{array} $$<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ F(x) = P(X \\le x) $$<\/p>\n<p>(a)<\/p>\n<p>$$ \\begin{align*} F(2) &amp; = P(X \\le 2) = 0.5 \\\\ 0.5 &amp; = P(X = 1) + P(X = 2) \\\\ &amp; = 0.2 + P(X = 2)\\\\ P(X = 2)&amp; = 0.5 \u2013 0.2 = 0.3 \\\\ \\end{align*} $$<\/p>\n<p>Note: a simpler, more direct approach can be:<\/p>\n<p>$$ P(X = 2) = F(2) \u2013 F(3) $$<\/p>\n<p>Therefore,<\/p>\n<p>$$ P(X = 2) = 0.5 \u2013 0.2 = 0.3 $$<\/p>\n<p>(b)<\/p>\n<p>$$ \\begin{align*} P(X = 4) &amp; = F(4) \u2013 F(3) \\\\ &amp;= 1 \u2013 0.85 = 0.15 \\\\ \\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>Given the following cumulative probability distribution, determine P(2).<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\text{Outcome} &amp; {0} &amp; {1} &amp; {2} &amp; {3} \\\\ \\hline \\text{Cumulative prob.} &amp; {1\/8} &amp; {4\/8} &amp; {7\/8} &amp; {1} \\\\ \\end{array} $$<\/p>\n<p>A. 7\/8<\/p>\n<p>B. 3\/8<\/p>\n<p>C. 1\/8<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is B.<\/p>\n<p>$$ \\begin{align*} P(2) &amp; = P(X = 2) \\\\ &amp; = F(2) \u2013 F(1) \\\\ &amp;=\\cfrac {7}{8} \u2013 \\cfrac {4}{8} \\\\ &amp; =\\cfrac {3}{8} \\\\ \\end{align*} $$<\/p>\n<\/blockquote>\n<p><em>Reading 9 LOS 9d:<\/em><\/p>\n<p><em>Calculate and interpret probabilities for a random variable, given its cumulative distribution function.<\/em><\/p>\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 offers a convenient tool for determining probabilities for a given random variable. As you have already learnt in a previous learning outcome statement, a cumulative distribution function, F(x), gives the probability that the random variable X is&#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-942","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>Calculating Probabilities using CDFs Example | CFA Level I - AnalystPrep<\/title>\n<meta name=\"description\" content=\"A cumulative distribution offers a convenient tool for determining probabilities for a given random variable which can easily be interpreted\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-probabilities-given-cumulative-distribution-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Calculating Probabilities using CDFs Example | CFA Level I - 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