{"id":771,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=771"},"modified":"2025-05-29T14:37:46","modified_gmt":"2025-05-29T14:37:46","slug":"discrete-random-variable-outcomes-example","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/","title":{"rendered":"Discrete Random Variables and Outcomes"},"content":{"rendered":"\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 session introduces probability distributions, distinguishing discrete and continuous variables, probability functions, cumulative distribution functions, and key distributions like normal and binomial. It covers statistical tools such as confidence intervals, Z-scores, Monte Carlo simulation, and applications in finance including risk assessment and portfolio management.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.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<h2>[vsw id=&#8221;KbEfz3KiJDo&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;]<\/h2>\n<p>A discrete random variable can take on a finite or countable number of values. It is a random variable where it is possible to list all the <a href=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/introduction-probability-defining-basic-terms\/\">outcomes<\/a>. Recall that a random variable is just a quantity whose future outcomes are not known with certainty. The shorthand notation for a discrete random variable is P(x) = P(X = x).<\/p>\n<p><!--more--><\/p>\n<h3><strong>Example 1: Discrete Random Variable Outcomes<\/strong><\/h3>\n<p>Consider a dividend with 7 possible year end values: $3, $5, $6, $9, $13, $14, and $15.<\/p>\n<p>Each of these is known as an outcome. Therefore, the set of possible outcomes can be expressed as:<\/p>\n<p>Div{$3, $5, $6, $9, $13, $14, $15}<\/p>\n<h3><strong>Example 2: Discrete Random Variable Outcomes<\/strong><\/h3>\n<p>Suppose we flipped a coin three times. What are the possible outcomes, and what is the probability of each?<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The best approach to such a question may involve a probability tree:<\/p>\n<p>$$<br \/>\n\\begin{array}{}<br \/>\n\\text{Probability Tree} \\\\<br \/>\n\\end{array}<br \/>\n$$<br \/>\n$$<br \/>\n\\begin{array}<br \/>\n\\hline<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{H} &amp; \\text{HHH} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{H} &amp; \\text{HH} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{T} &amp; \\text{HHT} \\\\<br \/>\n{} &amp; {} &amp; \\text{H} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{H} &amp; \\text{HTH} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{T} &amp; \\text{HH} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{T} &amp; \\text{HTT} \\\\<br \/>\n\\text{Start} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{H} &amp; \\text{THH} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{H} &amp; \\text{HH} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{T} &amp; \\text{THT} \\\\<br \/>\n{} &amp; {} &amp; \\text{T} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{H} &amp; \\text{TTH} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{T} &amp; \\text{HH} &amp; \\Huge \\begin{matrix} \\diagup \\\\ \\diagdown \\end{matrix} &amp; {} \\\\<br \/>\n{} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; {} &amp; \\text{T} &amp; \\text{TTT} \\\\<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>From the probability tree, we can get all the possible outcomes. Given that the probability of a head equals that of a tail:<\/p>\n<p>$$ P(H) = P(T) = 0.5 $$<\/p>\n<p>Suppose the random variable X represents the number of heads observed. The probability of not flipping heads at all is simply the number of outcomes without a head divided by the total number of outcomes. Therefore:<\/p>\n<p>$$ P(X = 0) =\\cfrac {1}{8} = 12.5\\%. $$<\/p>\n<p>Interpretation: in 12.5% of all trials, we can expect that heads will not be flipped.<\/p>\n<p>Similarly, the probability of flipping just one head is the number of outcomes with only one head divided by the total number of outcomes. Therefore:<\/p>\n<p>$$ P(X = 1) = \\cfrac {3}{8} $$<\/p>\n<p>We can calculate P(X = 2) and P(X = 3) following a similar technique.<\/p>\n<p>Finally, we can summarize the information in the form of a probability distribution as follows:<\/p>\n<p>$$<br \/>\n\\begin{array}{l|c|c|c|c}<br \/>\n\\textbf{Heads(outcomes)} &amp; \\text{0} &amp; \\text{1} &amp; \\text{2} &amp; \\text{3} \\\\<br \/>\n\\hline<br \/>\n\\textbf{Probability} &amp; \\text{1\/8} &amp; \\text{3\/8} &amp; \\text{3\/8} &amp; \\text{1\/8} \\\\<br \/>\n\\end{array}<br \/>\n$$<\/p>\n<p>Note that \\(\\sum P(X = x) = 1\\).<\/p>\n<p>In some instances, discrete variables may be treated as continuous variables, especially when the number of possible outcomes is quite large. For example, stock price movements on most exchanges are quoted in dollars and cents. However, the probability of a change of exactly $1.22 or $1.25 is very low (essentially zero). Such probabilities may not have <strong>material significance<\/strong>. For this reason, analysts may prefer working with price ranges. For instance:<\/p>\n<p>$$ P($2 \\le \\text{ price change } \\le $3) $$<\/p>\n<div class=\"notes_inv\">\n<hr \/>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>[vsw id=&#8221;KbEfz3KiJDo&#8221; source=&#8221;youtube&#8221; width=&#8221;611&#8243; height=&#8221;344&#8243; autoplay=&#8221;no&#8221;] A discrete random variable can take on a finite or countable number of values. It is a random variable where it is possible to list all the outcomes. Recall that a random variable 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-771","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>Discrete Random Variable Outcomes | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore examples of discrete random variable outcomes. Learn how to identify and list countable values for CFA Level 1 quantitative methods preparation.\" \/>\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\/discrete-random-variable-outcomes-example\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Discrete Random Variable Outcomes | CFA Level 1\" \/>\n<meta property=\"og:description\" content=\"Explore examples of discrete random variable outcomes. Learn how to identify and list countable values for CFA Level 1 quantitative methods preparation.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/\" \/>\n<meta property=\"og:site_name\" content=\"AnalystPrep | CFA\u00ae Exam Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2019-10-10T20:04:00+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-05-29T14:37:46+00:00\" \/>\n<meta name=\"author\" content=\"Ngari Joseph\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Ngari Joseph\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/\"},\"author\":{\"name\":\"Ngari Joseph\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\"},\"headline\":\"Discrete Random Variables and Outcomes\",\"datePublished\":\"2019-10-10T20:04:00+00:00\",\"dateModified\":\"2025-05-29T14:37:46+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/\"},\"wordCount\":638,\"articleSection\":[\"Quantitative Methods\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/\",\"name\":\"Discrete Random Variable Outcomes | CFA Level 1\",\"isPartOf\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\"},\"datePublished\":\"2019-10-10T20:04:00+00:00\",\"dateModified\":\"2025-05-29T14:37:46+00:00\",\"author\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\"},\"description\":\"Explore examples of discrete random variable outcomes. Learn how to identify and list countable values for CFA Level 1 quantitative methods preparation.\",\"breadcrumb\":{\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Discrete Random Variables and Outcomes\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#website\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/\",\"name\":\"AnalystPrep | CFA\u00ae Exam Study Notes\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914\",\"name\":\"Ngari Joseph\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png\",\"contentUrl\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png\",\"caption\":\"Ngari Joseph\"},\"url\":\"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/ngari-joseph\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Discrete Random Variable Outcomes | CFA Level 1","description":"Explore examples of discrete random variable outcomes. Learn how to identify and list countable values for CFA Level 1 quantitative methods preparation.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/","og_locale":"en_US","og_type":"article","og_title":"Discrete Random Variable Outcomes | CFA Level 1","og_description":"Explore examples of discrete random variable outcomes. Learn how to identify and list countable values for CFA Level 1 quantitative methods preparation.","og_url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/","og_site_name":"AnalystPrep | CFA\u00ae Exam Study Notes","article_published_time":"2019-10-10T20:04:00+00:00","article_modified_time":"2025-05-29T14:37:46+00:00","author":"Ngari Joseph","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Ngari Joseph","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/"},"author":{"name":"Ngari Joseph","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914"},"headline":"Discrete Random Variables and Outcomes","datePublished":"2019-10-10T20:04:00+00:00","dateModified":"2025-05-29T14:37:46+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/"},"wordCount":638,"articleSection":["Quantitative Methods"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/","name":"Discrete Random Variable Outcomes | CFA Level 1","isPartOf":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website"},"datePublished":"2019-10-10T20:04:00+00:00","dateModified":"2025-05-29T14:37:46+00:00","author":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914"},"description":"Explore examples of discrete random variable outcomes. Learn how to identify and list countable values for CFA Level 1 quantitative methods preparation.","breadcrumb":{"@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/discrete-random-variable-outcomes-example\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/cfa-level-1-exam\/"},{"@type":"ListItem","position":2,"name":"Discrete Random Variables and Outcomes"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#website","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/","name":"AnalystPrep | CFA\u00ae Exam Study Notes","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/cfa-level-1-exam\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/ddbf15286b86900dfa9f8ac2603c3914","name":"Ngari Joseph","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/analystprep.com\/cfa-level-1-exam\/#\/schema\/person\/image\/","url":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png","contentUrl":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2016\/08\/male3-512-150x150.png","caption":"Ngari Joseph"},"url":"https:\/\/analystprep.com\/cfa-level-1-exam\/author\/ngari-joseph\/"}]}},"_links":{"self":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/771","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/comments?post=771"}],"version-history":[{"count":25,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/771\/revisions"}],"predecessor-version":[{"id":56288,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/posts\/771\/revisions\/56288"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/media?parent=771"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/categories?post=771"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-json\/wp\/v2\/tags?post=771"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}