{"id":29629,"date":"2021-09-08T10:58:40","date_gmt":"2021-09-08T10:58:40","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29629"},"modified":"2026-03-11T07:30:11","modified_gmt":"2026-03-11T07:30:11","slug":"calculating-probabilities-from-cumulative-distribution-function","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/calculating-probabilities-from-cumulative-distribution-function\/","title":{"rendered":"Calculating Probabilities from Cumulative Distribution Function"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Common Probability Distributions (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 Module 4)\",\n  \"description\": \"CFA\u00ae Level I Quantitative Methods video lesson by AnalystPrep covering Common Probability Distributions (Module 4). This lecture explains probability distributions, discrete versus continuous random variables, cumulative distribution functions, discrete and continuous uniform distributions, Bernoulli and binomial distributions, properties of the normal distribution, univariate versus multivariate distributions, standardization, probabilities using the standard normal distribution, shortfall risk and Roy\u2019s safety-first criterion, lognormal distributions, continuously compounded returns, Student\u2019s t-, chi-square, and F-distributions, and an introduction to Monte Carlo simulation, with exam-focused examples.\",\n  \"uploadDate\": \"2021-11-09T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/TXO9ODcLWiU\/default.jpg\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=TXO9ODcLWiU\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/TXO9ODcLWiU\",\n  \"duration\": \"PT1H47S\"\n}\n<\/script>\n\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(X = 2).\",\n    \"text\": \"Given the following cumulative probability distribution, determine P(X = 2).\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"3\/8.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"7\/8.\"\n      },\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"1\/8.\"\n      }\n    ]\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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&#8221; to the specified value. Perhaps an example will make this concept clearer.<\/p>\n<div style=\"text-align: center; margin: 20px 0;\"><a style=\"display: inline-block; padding: 10px 18px; border: 2px solid #1e5bd8; color: #1e5bd8; border-radius: 9999px; text-decoration: none; font-weight: 600;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Practice CDF probability questions for CFA Level I <\/a><\/div>\n<h4><strong>Example: Cumulative Distribution<\/strong><\/h4>\n<p>If we flipped a coin three times, we would end up with the following probability distribution of the number of heads obtained:<\/p>\n<p>$$ \\begin{array}{c|c|c|c} \\textbf{Heads (outcomes)} &amp; \\bf{0} &amp; \\bf{1} &amp; \\bf{2} &amp; \\bf{3} \\\\ \\hline \\text{Probability} &amp; {1\/8} &amp; {3\/8} &amp; {3\/8} &amp; {1\/8} \\\\ \\end{array} $$<\/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>$$ \\begin{array}{c|c|c|c} \\textbf{Heads (outcomes)} &amp; \\bf{0} &amp; \\bf{1} &amp; \\bf{2} &amp; \\bf{3} \\\\ \\hline \\text{Probability} &amp; {1\/8} &amp; {3\/8} &amp; {3\/8} &amp; {1\/8} \\\\ \\hline \\text{Cumulative prob.} &amp; {1\/8} &amp; {4\/8} &amp; {7\/8} &amp; {8\/8} \\\\ \\end{array} $$<\/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.<\/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. The probability of at most two heads from the cumulative distribution above is 0.875.<\/p>\n<h4><strong>Example: Cumulative Distribution Function<\/strong><\/h4>\n<p>Variable X can take values 1, 2, 3, and 4. The probability of each outcome has been given below.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\textbf{Outcome} &amp; \\bf{1} &amp; \\bf{2} &amp; \\bf{3} &amp; \\bf{4} \\\\ \\hline \\text{Probability} &amp; {0.2} &amp; {0.3} &amp; {0.35} &amp; {0.15} \\\\ \\end{array} $$<\/p>\n<p>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>$$ \\begin{array}{c|c|c|c|c} \\textbf{Heads (outcomes)} &amp; \\bf{1} &amp; \\bf{2} &amp;\\bf {3} &amp; \\bf{4} \\\\ \\hline \\text{Probability} &amp; {0.2} &amp; {0.3} &amp; {0.35} &amp; {0.15} \\\\ \\hline \\text{Cumulative prob.} &amp; {0.2} &amp; {0.5} &amp; {0.85} &amp; {1} \\\\ \\end{array} $$<\/p>\n<p>From the table, it is clear that \\(P(X \\le 2) = 0.5\\).<\/p>\n<p><em><strong>Note to candidates<\/strong><\/em>: The standard notation for a cumulative distribution function is written in upper case \\(F(x)\\). In contrast, that of a probability function is written in lowercase \\(f(x)\\).<\/p>\n<h2><strong>Calculating Probabilities Given Cumulative Distribution Function<\/strong><\/h2>\n<div class=\"notes_inv\">\n<p>A cumulative distribution offers a convenient tool for determining probabilities for a given random variable. As seen above, the 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><!--more--><\/p>\n<p>$$ F(x) = P(X \\le x) $$<\/p>\n<h4><strong>Example: Cumulative Distribution Function<\/strong><\/h4>\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)\\).<\/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 probability distribution function (pdf).<\/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>$$\\begin{align} P(20) &amp;=\\cfrac {20}{150} \\\\ P(30) &amp;=\\cfrac {30}{150}\\\\ P(40) &amp;=\\cfrac {40}{150}\\end{align}$$<\/p>\n<p>And lastly,<\/p>\n<p>$$ P(50) =\\cfrac {50}{150} $$<\/p>\n<p><strong><em>Note to candidates<\/em><\/strong>: We can prove that our pdf is correct by testing the first rule of probability distribution functions by adding all the probabilities.<\/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><em><strong>Interpretation<\/strong><\/em>: 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><strong>Interpretation<\/strong>: There is a 66.67% cumulative probability that outcomes 10, 20, 30, or 40 occur.<\/p>\n<h4><strong>Example: Calculating Probabilities Given Cumulative Distribution Function<\/strong><\/h4>\n<p>Variable \\(X\\) can take the values 1, 2, 3, and 4. The cumulative probability distribution is given below. Use it to calculate:<\/p>\n<p>(a) P(X = 2).<\/p>\n<p>(b) P(X = 4).<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\textbf{Outcome} &amp; \\bf{1} &amp; \\bf{2} &amp; \\bf{3} &amp; \\bf{4} \\\\ \\hline \\text{Cumulative Probability Distribution} &amp; {0.2} &amp; {0.5} &amp; {0.85} &amp; {1} \\\\ \\end{array} $$<\/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><em><strong>Note to candidates<\/strong><\/em>: 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(X=2)\\).<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c} \\textbf{Outcome} &amp; \\bf{0} &amp; \\bf{1} &amp; \\bf{2} &amp; \\bf{3} \\\\ \\hline \\text{Cumulative prob.} &amp; {1\/8} &amp; {4\/8} &amp; {7\/8} &amp; {1} \\\\ \\end{array} $$<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>\\(\\frac{7}{8}\\).<\/li>\n<li>\\(\\frac{3}{8}\\).<\/li>\n<li>\\(\\frac{1}{8}\\).<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>$$ \\begin{align*}\u00a0 P(X = 2)&amp; = F(2) \u2013 F(1) \\\\ &amp;=\\cfrac {7}{8} \u2013 \\cfrac {4}{8} \\\\ &amp; =\\cfrac {3}{8} \\\\ \\end{align*} $$<\/p>\n<\/blockquote>\n<div style=\"text-align: center; margin: 30px 0;\"><a style=\"display: inline-block; padding: 12px 24px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"> Start Free Trial \u2192 <\/a>\n<p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">Strengthen your CFA Level I quantitative methods skills with exam-style practice on probability distributions, cumulative distribution functions, and probability calculations.<\/p>\n<\/div>\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":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-29629","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 from CDF | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn to calculate probabilities using the cumulative distribution function (CDF). 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