{"id":945,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=945"},"modified":"2025-12-15T16:22:34","modified_gmt":"2025-12-15T16:22:34","slug":"definition-discrete-uniform-random-variable-bernoulli-random-variable-binomial-random-variable","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/definition-discrete-uniform-random-variable-bernoulli-random-variable-binomial-random-variable\/","title":{"rendered":"Discrete Uniform Random Variables, a Bernoulli Random Variables, and Binomial Random Variables"},"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 video lesson covers common probability distributions and their applications in quantitative methods. It explains key concepts such as discrete vs. continuous random variables, probability functions, cumulative distribution functions, Bernoulli trials, binomial and uniform distributions, normal distribution properties, confidence intervals, Monte Carlo simulation, and their relevance to financial analysis and risk management.\",\n  \"uploadDate\": \"2019-12-18T00:00:00+00:00\",\n  \"thumbnailUrl\": \"https:\/\/i.ytimg.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<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?si=23I0NKqck3mTsQGQ\" 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<h2>\u00a0<\/h2>\n<p>Probability distributions have different shapes and characteristics. As such, we describe a random variable based on the shape of the underlying distribution.<\/p>\n<p><!--more--><\/p>\n<h2><strong>A Discrete Uniform Random Variable<\/strong><\/h2>\n<p>A discrete uniform random variable is one whose probabilities for all possible outcomes are equal. For example, if we throw a die, the probability of any value between 1 and 6 is 1\/6. Therefore, the throw of a die is a uniform distribution with a discrete random variable.<\/p>\n<p>In finance, uniform discrete random variables are usually used in simulations. In such instances, financial managers might be interested in drawing a random number such that each random number within a given range has the same probability of being selected.<\/p>\n<h2><strong>A Bernoulli Random Variable<\/strong><\/h2>\n<p>A Bernoulli trial is an experiment that has only two outcomes: success (S) or failure (F). You should note that we use the words \u201csuccess\u201d and \u201cfailure\u201d just for labeling purposes. These words, therefore, may not necessarily carry with them their denotative meanings.<\/p>\n<p>We define the random variable X by X(s) = 1, X(f) = 0. As such, X is the number of successes that occur (0 or 1).<\/p>\n<p>A Bernoulli variable can sometimes be used as an \u201cindicator\u201d to indicate the occurrence of a given event. We could set X = 1 if event B occurs and X = 0 if event B does not occur. For example, event B could be a return of over 10% on a stock.<\/p>\n<h2><strong>A Binomial Random Variable<\/strong><\/h2>\n<p>A binomial random variable is the number of successes in n Bernoulli trials where:<\/p>\n<ol>\n<li>the trials are independent \u2013 the outcome of any trial does not depend on the outcomes of the other trials;<\/li>\n<li>the trials are identical (the probability of success is equal for all trials).<\/li>\n<\/ol>\n<p>For example, the tossing of a coin has two mutually exclusive outcomes, where the probability of the outcome of any toss (trial) is not affected by outcomes of prior trials.<\/p>\n<div class=\"notes_inv\"><hr \/><\/div>","protected":false},"excerpt":{"rendered":"<p>\u00a0 Probability distributions have different shapes and characteristics. As such, we describe a random variable based on the shape of the underlying distribution.<\/p>\n","protected":false},"author":24,"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-945","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 Uniform &amp; Binomial Variables | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn the definitions and examples of discrete uniform, Bernoulli, and binomial random variables. 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