{"id":1228,"date":"2019-10-10T20:04:00","date_gmt":"2019-10-10T20:04:00","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=1228"},"modified":"2026-01-09T12:40:53","modified_gmt":"2026-01-09T12:40:53","slug":"univariate-vs-multivariate-distribution","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/univariate-vs-multivariate-distribution\/","title":{"rendered":"Univariate vs. Multivariate Distributions and the Role of Correlation in the Multivariate Normal Distribution"},"content":{"rendered":"\n<iframe loading=\"lazy\"\n  width=\"611\"\n  height=\"344\"\n  src=\"https:\/\/www.youtube.com\/embed\/KbEfz3KiJDo?rel=0&#038;modestbranding=1&#038;playsinline=1\"\n  title=\"YouTube video player\"\n  frameborder=\"0\"\n  allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\"\n  allowfullscreen>\n<\/iframe>\n\n\n\n<h2>\u00a0<\/h2>\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 regard, the strength of the relationship between the variables (correlation) is very important. As you will recall, a linear combination of 2 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 on a group of assets. Correlation defines the strength of the linear relationship between any 2 random variables. For us to define a multivariate distribution (n variables), we need the following:<\/p>\n<ol>\n<li>the individual mean value for each asset (\u03bc<sub>1<\/sub>, \u03bc<sub>2<\/sub>, \u03bc<sub>3<\/sub>, \u2026, \u03bc<sub>n-1<\/sub>, \u03bc<sub>n<\/sub>);<\/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, \\sigma_{n-1}^2)\\); and<\/li>\n<li>pairwise return correlations \u2013 \\(\\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<h3><strong>Example:\u00a0<\/strong><strong>Multivariate Normal Distributions<\/strong><\/h3>\n<p>Suppose we wish to model the distribution of two asset returns: to describe the return multivariate distribution, we will need two means, two variances, and just one correlation &#8211; \\(\\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 &#8211; \\(\\frac {5(5 \u2013 1)}{2} = 10.\\)<\/p>\n<p>In conclusion, if we have individual security returns that are jointly, normally distributed, the return of a portfolio made up of such assets will also be normally distributed. As such, we can come up with a multivariate distribution provided we have the three items that we have listed above.<\/p>\n<div class=\"notes_inv\"><hr \/><\/div>","protected":false},"excerpt":{"rendered":"<p>\u00a0 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":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-1228","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 Distributions | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn the differences between univariate and multivariate distributions, including their probability functions and applications.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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