{"id":27496,"date":"2021-08-25T06:03:29","date_gmt":"2021-08-25T06:03:29","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=27496"},"modified":"2026-03-02T11:05:57","modified_gmt":"2026-03-02T11:05:57","slug":"expected-value-variance-and-standard-deviation-of-random-variables","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/expected-value-variance-and-standard-deviation-of-random-variables\/","title":{"rendered":"Expected Value, Variance, and Standard Deviation of Random Variables"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the standard deviation of expected returns given probabilities?\",\n    \"text\": \"You are given the following returns and corresponding probabilities:\\n\\nReturn | Probability\\n4% | 40%\\n5% | 25%\\n6% | 35%\\n\\nWhat is the standard deviation of expected returns closest to?\\n\\nA. 0.00007475.\\n\\nB. 0.0495.\\n\\nC. 0.008646.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is C. First compute the expected return: E(X) = (0.04 \u00d7 0.40) + (0.05 \u00d7 0.25) + (0.06 \u00d7 0.35) = 0.0495. Next compute the variance: 0.4(0.04 \u2212 0.0495)^2 + 0.25(0.05 \u2212 0.0495)^2 + 0.35(0.06 \u2212 0.0495)^2 = 0.00007475. Taking the square root gives the standard deviation: \u221a0.00007475 = 0.008646.\"\n    }\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/PsZrsg3ZUDE\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Expected Value<\/strong><\/h2>\n<p>The expected value of a random variable is the average of the possible outcomes of that variable, taking the probability weights into account. Therefore:<\/p>\n<p>$$ E\\left( X \\right) =\\sum _{ i=1 }^{ n }{ { X }_{ i }P\\left( { X }_{ i } \\right) } $$<\/p>\n<h4><strong>Example: Expected Value<\/strong><\/h4>\n<p>An analyst anticipates the following returns from an asset:<\/p>\n<p>$$ \\begin{array}{c|c} {\\textbf{Return}} &amp; {\\textbf{Probability}} \\\\ \\hline {5\\%} &amp; { 65\\%} \\\\ \\hline {7\\%} &amp; { 25\\%} \\\\ \\hline {8\\%} &amp; { 10\\%} \\\\ \\end{array} $$<\/p>\n<p>The expected value of the investment is <em>closest<\/em> to:<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$ \\begin{align*} \\text{Expected return} &amp; = 0.05 \u00d7 0.65 + 0.07 \u00d7 0.25 + 0.10 \u00d7 0.08 \\\\ &amp; = 0.0325 + 0.0175 + 0.008 \\\\ &amp; = 0.058 \\\\ \\end{align*} $$<\/p>\n<h2><strong>Variance<\/strong><\/h2>\n<p>The variance of a random variable is the sum of the squared deviations from the expected value weighted by respective probabilities. Therefore:<\/p>\n<p>$$ { \\sigma }^{ 2 }\\left( X \\right) =\\sum _{ i=1 }^{ n }{ { \\left[ { X }_{ i }-E\\left( { X } \\right) \\right] }^{ 2 }P } \\left( { X }_{ i } \\right) =\\left\\{ { \\left[ X-E\\left( { X } \\right) \\right] }^{ 2 } \\right\\} $$<\/p>\n<h4><strong>Example: Calculating Variance<\/strong><\/h4>\n<p>Using the data from the previous example, we can compute the variance of return:<\/p>\n<p>$$ \\begin{align*} { \\sigma }^{ 2 }\\left( X \\right) &amp; =0.65{ (0.05-0.058) }^{ 2 }+0.25{ (0.07-0.058) }^{ 2 }+0.10{ (0.08-0.058) }^{ 2 } \\\\ &amp; = 0.000126 \\\\ \\end{align*} $$<\/p>\n<h2><strong>Standard Deviation<\/strong><\/h2>\n<p>Variance is not easy to interpret because it has squared units. Therefore, we usually use the standard deviation, which has the same units as the expected value. To get the standard deviation, we use the square root of variance:<\/p>\n<p>$$ \\begin{align*} \\text{Standard deviation} &amp; = \\sqrt{\\text{Variance}} \\\\ &amp;= \\sqrt{0.000126} \\\\ &amp; =0.01122 \\text{ or } 1.12\\% \\\\ \\end{align*} $$<\/p>\n<p><em><strong>Note:<\/strong><\/em> You can always raise the variance to 0.5 power to get the same result.<\/p>\n<p>$$ \\begin{align*} \\text{Standard deviation} &amp; = \\text{Variance}^{0.5} \\\\ &amp;= 0.000126^{0.5} \\\\ &amp; =0.01122 \\text{ or } 1.12\\% \\\\ \\end{align*} $$<\/p>\n<blockquote>\n<h2><strong>Question<\/strong><\/h2>\n<p>You have been given the following data indicating the returns likely to be earned on a stock alongside the corresponding probabilities:<\/p>\n<p>$$ \\begin{array}{c|c} {\\textbf{Return}} &amp; {\\textbf{Probability}} \\\\ \\hline {4\\%} &amp; { 40\\%} \\\\ \\hline {5\\%} &amp; { 25\\%} \\\\ \\hline {6\\%} &amp; { 35\\%} \\\\ \\end{array} $$<\/p>\n<p>The standard deviation of expected returns is <em>closest<\/em> to:<\/p>\n<p>A. 0.00007475.<\/p>\n<p>B. 0.0495.<\/p>\n<p>C. 0.008646.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>C<\/strong>.<\/p>\n<p>The first step involves determining the expected return:<\/p>\n<p>$$ \\begin{align*} E(X) &amp; = (0.04 \u00d7 0.4) + (0.05 \u00d7 0.25) + (0.06 \u00d7 0.35) \\\\ &amp; = 0.0495 \\\\ \\end{align*} $$<\/p>\n<p>Next, we must compute the variance of returns:<\/p>\n<p>$$ \\begin{align*} { \\sigma }^{ 2 }\\left( X \\right) &amp; =0.4(0.04\u20130.0495)^{ 2 }+0.25(0.05\u20130.0495)^{ 2 }+0.35(0.06 \u2013 0.0495)^{ 2 } \\\\ &amp; = 0.00007475 \\\\ \\end{align*} $$<\/p>\n<p>Lastly, we find the square root of variance to get the standard deviation of expected return:<\/p>\n<p>$$ { \\sigma }= \\sqrt{0.00007475} = 0.008646 $$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Expected Value The expected value of a random variable is the average of the possible outcomes of that variable, taking the probability weights into account. Therefore: $$ E\\left( X \\right) =\\sum _{ i=1 }^{ n }{ { X }_{ i&#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-27496","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>Expected Value, Variance &amp; Standard Deviation | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate expected value, variance, and standard deviation of random variables using probability-weighted outcomes.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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