{"id":29619,"date":"2021-09-08T05:53:40","date_gmt":"2021-09-08T05:53:40","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29619"},"modified":"2025-12-12T12:29:39","modified_gmt":"2025-12-12T12:29:39","slug":"29619","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/29619\/","title":{"rendered":"Probability Distribution of Discrete and Continous 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\": \"Which of the following is least likely a property of probability mass function, P(x)?\",\n    \"text\": \"A. The probability mass function is nonnegative for all x.\\nB. The integral of the probability mass function is 1.\\nC. The sum of the probabilities of all possible outcomes = 1.\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is B.\\n\\nOption B is a property of probability density function (for continuous random variables) and not probability mass function.\"\n    },\n    \"suggestedAnswer\": [\n      {\n        \"@type\": \"Answer\",\n        \"text\": \"A and C are incorrect. For a discrete random variable, the probability function is termed as probability mass function with the following properties:\\nP(X = x) = P(x). In statistics, P(x) is said to be a probability mass function.\\nP(x) is always nonnegative for all x.\\nThe sum of the probabilities of all possible outcomes = 1.\"\n      }\n    ]\n  }\n}\n<\/script>\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-104.jpg\",\n  \"caption\": \"Probability Distribution\",\n  \"width\": 1613,\n  \"height\": 964,\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\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/TXO9ODcLWiU\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Probability Distribution<\/strong><\/h2>\n<p>The probability distribution of a random variable \\(X\\) is a <strong>graphical presentation <\/strong>of the probabilities associated with the possible outcomes of \\(X\\). A random variable is any quantity for which more than one value is possible. The price of quoted stocks is a good example in this respect. Simply put, a probability distribution gathers all the outcomes and further indicates the probability associated with each outcome.<\/p>\n<p>This reading covers the probability distributions listed below:<\/p>\n<ul>\n<li>Uniform.<\/li>\n<li>Binomial.<\/li>\n<li>Normal.<\/li>\n<li>Lognormal.<\/li>\n<li>Student\u2019s t.<\/li>\n<li>Chi-square.<\/li>\n<li>F-distribution.<\/li>\n<\/ul>\n<h4><strong>Example: Probability Distribution<\/strong><\/h4>\n<p>Suppose we roll a die. The set of possible outcomes is:<\/p>\n<p>{ 1\u00a0\u00a0 2\u00a0\u00a0 3\u00a0\u00a0 4\u00a0\u00a0 5\u00a0\u00a0 6 }<\/p>\n<p>Each of these outcomes would occur with a probability of 1\/6 because each outcome has an equal chance of occurrence. Consequently, the probability distribution would be a straight line:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-16968\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-104.jpg\" alt=\"\" width=\"1613\" height=\"964\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-104.jpg 1613w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-104-300x179.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-104-768x459.jpg 768w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-104-1024x612.jpg 1024w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2019\/10\/page-104-400x239.jpg 400w\" sizes=\"auto, (max-width: 1613px) 100vw, 1613px\" \/><\/p>\n<p><em><strong>Note to candidates<\/strong><\/em>: Although the above distribution is a straight line, most real-life distributions are usually curved. We will particularly delve into the <strong>bell-shaped normal distribution<\/strong> later.<\/p>\n<h2><strong>Discrete Random Variables<\/strong><\/h2>\n<p>A discrete random variable can take on a finite number of outcomes. Examples include:<\/p>\n<ol style=\"list-style-type: lower-roman;\">\n<li>If we roll a die, there are 6 possible outcomes. Therefore, the outcomes are discrete and random.<\/li>\n<li>The number of CFA\u00ae charterholders employed within a given year is a discrete random variable.<\/li>\n<\/ol>\n<h2><strong>Continuous Random Variables<\/strong><\/h2>\n<p>A continuous random variable is one that has an infinite number of possible outcomes. A good example can be the rate of return on a stock. For instance, the return can be 6%, or between 6% and 7%, in which case, it can take on 6.4%, 6.41%, 6.412%, or even 6.412325%, i.e., infinite values.<\/p>\n<h2><strong>Probability Function<\/strong><\/h2>\n<p>A probability function gives the probability of a random variable \\(X\\) taking on a value \u201c\\(x\\).\u201d The probability functions of discrete and continuous random variables are slightly different.<\/p>\n<p>For a discrete random variable, the probability function, \\(P(x)\\), satisfies the following properties:<\/p>\n<ol>\n<li>\\(P(X = x) = P(x)\\).<br \/>\nIn statistics, \\(P(x)\\) is said to be a probability mass function.<\/li>\n<li>\\(P(x)\\) is always nonnegative for all \\(x\\).<\/li>\n<li>The sum of the probabilities of all possible outcomes = 1.<\/li>\n<\/ol>\n<p>For a continuous random variable, the probability function, \\(f(x)\\), satisfies the following properties:<\/p>\n<ol>\n<li>The probability that \\(x\\) is between two values, \\(a\\) and \\(b\\) is:<br \/>\n$$ P\\left\\{ a\\le x\\le b \\right\\} =\\int _{ a }^{ b }{ f\\left( x \\right) dx } $$<br \/>\n\\(f(x)\\) is said to be a probability density function (pdf).<\/li>\n<li>\\(f(x)\\) is nonnegative for all \\(x\\).<\/li>\n<li>The integral of the probability function is 1, that is,<br \/>\n$$ \\int _{ -\\infty }^{ \\infty }{ f\\left( x \\right) dx=1 } $$<\/li>\n<\/ol>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Which of the following is\u00a0<em>least likely a <\/em>property of probability mass function, P(x)?<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>The probability mass function is nonnegative for all \\(x\\).<\/li>\n<li>The integral of the probability mass function is 1.<\/li>\n<li>The sum of the probabilities of all possible outcomes = 1.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is <strong>B<\/strong>.<\/p>\n<p>Option B is a property of probability density function (for continuous random variables) and not probability mass function.<\/p>\n<p><strong>A and C are incorrect<\/strong>. For a discrete random variable, the probability function is termed as probability mass function with the following properties:<\/p>\n<ol>\n<li>\\(P(X = x) = P(x)\\).<br \/>\nIn statistics, \\(P(x)\\) is said to be a probability mass function.<\/li>\n<li>\\(P(x)\\) is always nonnegative for all \\(x\\).<\/li>\n<li>The sum of the probabilities of all possible outcomes = 1.<\/li>\n<\/ol>\n<\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Probability Distribution The probability distribution of a random variable \\(X\\) is a graphical presentation of the probabilities associated with the possible outcomes of \\(X\\). A random variable is any quantity for which more than one value is possible. The price&#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-29619","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Discrete vs. Continuous Probability | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn the differences between discrete and continuous probability distributions, including key concepts and graphical representations.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" 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