{"id":29800,"date":"2021-09-09T03:44:34","date_gmt":"2021-09-09T03:44:34","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=29800"},"modified":"2026-03-11T13:57:42","modified_gmt":"2026-03-11T13:57:42","slug":"univariate-distribution-multivariate-distribution-and-correlation","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/univariate-distribution-multivariate-distribution-and-correlation\/","title":{"rendered":"Univariate Distribution, Multivariate Distribution, and Correlation"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"How many distinct correlations are required to define a multivariate distribution for three asset returns?\",\n    \"text\": \"Suppose we wish to model the distribution of three asset returns so as to describe the return multivariate distribution. The number of distinct correlations needed is closest to:\\n\\nA. 6\\nB. 10\\nC. 40\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A. The number of distinct correlations required to define a multivariate distribution with n variables is n(n\u22121)\/2. For three asset returns, this equals 3(3\u22121)\/2 = 3. However, among the given options, 6 is the closest available choice.\"\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>Univariate and multivariate normal distributions are very robust and useful in most statistical procedures. Understanding their form and function will help you learn a lot about most statistical routines.<\/p>\n<p><!--more--><\/p>\n<h2><strong>Univariate Distributions<\/strong><\/h2>\n<p>A univariate distribution is defined as a distribution that involves just one random variable. For instance, suppose we wish to model the distribution of returns on an asset, such as a holding of stocks; such a model would be a univariate distribution.<\/p>\n<p>In previous learning outcome statements, we have been focusing on univariate distributions such as the binomial, uniform, and normal distributions. Let us now look at multivariate distributions.<\/p>\n<h2><strong>Multivariate Distributions<\/strong><\/h2>\n<p>A multivariate distribution describes the probabilities for a group of continuous random variables, particularly if the individual variables follow a normal distribution. Each variable has its own mean and variance. In this case, the strength of the relationship between the variables (correlation) is very important. As you will recall, a linear combination of two normal random variables results in another normal random variable.<\/p>\n<h2><strong>The Role of Correlation in Multivariate Normal Distributions<\/strong><\/h2>\n<p>We could be interested, for instance, in the distribution of returns in a group of assets. Correlation defines the strength of the linear relationship between any two random variables. For us to define a multivariate distribution with \\(n\\) variables, we need the following:<\/p>\n<ol style=\"list-style-type: lower-roman;\">\n<li>The individual mean value for each asset (\\(\u03bc_1, \u03bc_2, \u03bc_3, \u2026, \u03bc_n\\)).<\/li>\n<li>A list of variances of return for each asset (\\(\\sigma_1^2, \\sigma_2^2, \\sigma_3^2, \u2026, \\sigma_{n}^2\\).<\/li>\n<li>Pairwise return correlations, so \\(\\cfrac {n(n-1)}{2}\\) correlations in total.<\/li>\n<\/ol>\n<p>Correlation is the distinguishing feature between univariate and multivariate normal distributions.<\/p>\n<h4><strong>Example:\u00a0<\/strong><strong>Multivariate Normal Distributions<\/strong><\/h4>\n<p>Suppose we wish to model the distribution of two asset returns so as to describe the return multivariate distribution; we will need two means, two variances, and just one correlation, \\(\\frac {2(2 \u2013 1)}{2} = 1.\\)<\/p>\n<p>On the other hand, if we had 5 assets, we would need to establish 5 means, 5 variances, and 10 distinct correlation values, \\(\\frac {5(5 \u2013 1)}{2} = 10.\\)<\/p>\n<p>In conclusion, if we have individual security returns that are jointly distributed normally, the return of a portfolio made up of such assets will also be distributed normally. In our analysis of such a portfolio, we can come up with a multivariate distribution provided we have the three items that we have listed above (mean, variance, and correlation).<\/p>\n<blockquote>\n<h3><strong>Question<\/strong><\/h3>\n<p>Suppose we wish to model the distribution of three asset returns so as to describe the return multivariate distribution; the number of distinct correlations needed is <em>closest to:<\/em><\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>6.<\/li>\n<li>10.<\/li>\n<li>40.<\/li>\n<\/ol>\n<p><strong>Solution<\/strong><\/p>\n<p>The correct answer is\u00a0<strong>B<\/strong>.<\/p>\n<p>Recall that to define a multivariate distribution with \\(n\\) variables, we need:<\/p>\n<p>$$\\cfrac {n(n-1)}{2} \\ \\text{distinct correlation values}$$<\/p>\n<p>In this case we need:<\/p>\n<p>$$\\frac {3(3 \u2013 1)}{2} = 6\\ \\text{distinct correlation values}.$$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>Univariate and multivariate normal distributions are very robust and useful in most statistical procedures. Understanding their form and function will help you learn a lot about most statistical routines.<\/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-29800","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>Univariate vs. Multivariate Distribution | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Explore the differences between univariate and multivariate distributions, their applications, and correlation analysis in statistics.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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