{"id":2755,"date":"2014-07-18T03:32:02","date_gmt":"2014-07-18T03:32:02","guid":{"rendered":"http:\/\/www.rationalargumentator.com\/index\/?p=2755"},"modified":"2014-07-18T03:32:02","modified_gmt":"2014-07-18T03:32:02","slug":"uniform-distribution","status":"publish","type":"post","link":"https:\/\/www.rationalargumentator.com\/index\/blog\/2014\/07\/uniform-distribution\/","title":{"rendered":"Ideas in Mathematics and Probability: The Uniform Distribution (2007) – Article by G. Stolyarov II"},"content":{"rendered":"
\n
\"The<\/div>\n<\/div>\n
G. Stolyarov II
\n<\/big><\/big><\/strong><\/span><\/div>\n
July 17, 2014<\/span>
\n<\/big><\/div>\n
******************************<\/div>\n
Note from the Author: <\/strong>This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.\u00a0 The article earned over 4,800 page views on Associated Content\/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.\u00a0 ***<\/span><\/em><\/div>\n
***<\/span><\/div>\n
~ G. Stolyarov II, July 17, 2014<\/em><\/div>\n
\n
***<\/span><\/div>\n<\/div>\n

The uniform distribution is alternately known as the de Moivre distribution, in honor of the French mathematician Abraham de Moivre<\/a> (1667-1754) who introduced it to probability theory. The fundamental assumption behind the uniform distribution is that none of the possible outcomes is more or less likely than any other.<\/i> The uniform distribution applies to continuous random variables, i.e., variables that can assume any values within a specified range.***<\/span><\/p>\n

Let us say that a given random variable X is uniformly distributed over the interval from a to b. That is, the smallest value X can assume is a and the largest value it can assume is b. To determine the probability density function (pdf) of such a random variable, we need only remember that the total area under the graph of the pdf must equal 1. Since the pdf is constant throughout the interval on which X can assume values, the area underneath its graph is that of a rectangle — which can be determined by multiplying its base by its height. But we know the base of the rectangle to be (b-a), the width of the interval over which the random variable is distributed, and its area to be 1. Thus, the height of the rectangle must be 1\/(b-a)<\/b>, which is also the probability density function of a uniform random variable over the region from a to b.<\/p>\n

What is the mean of a uniformly distributed random variable? It is, conveniently, the halfway point of the interval from a to b, since half of the entire area under the graph of the pdf will be to the right of such a midway point, and half will be to the left. So the mean or mathematical expectation of a uniformly distributed random variable is (b-a)\/2<\/b>.<\/p>\n

It is also possible to arrive at a convenient formula for the variance of such a uniform variable. Let us consider the following equation used for determining variance:<\/p>\n

Var(X) = E(X2<\/sup>) – E(X)2<\/sup> , where X is our uniformly distributed random variable.<\/p>\n

We already know that E(X) = (b-a)\/2, so E(X)2<\/sup> must equal (b-a)2<\/sup>\/4. To find E(X2<\/sup>), we can use the definition of such an expectation as the definite integral of x2<\/sup>*f(x) evaluated from b to a, where f(x) is the pdf of our random variable. We already know that f(x) = 1\/(b-a); so E(X2<\/sup>) is equal to the integral of x2<\/sup>\/(b-a), or x3<\/sup>\/3(b-a), evaluated from b to a, which becomes (b-a)3<\/sup>\/3(b-a), or (b-a)2<\/sup>\/3.<\/p>\n

Thus, Var(X) = E(X2<\/sup>) – E(X)2<\/sup> = (b-a)2<\/sup>\/3 – (b-a)2<\/sup>\/4 = (b-a)2<\/sup>\/12<\/b>, which is the variance for any uniformly distributed random variable.<\/p>\n","protected":false},"excerpt":{"rendered":"

G. Stolyarov II July 17, 2014 ****************************** Note from the Author: This article was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007.\u00a0 The article earned over 4,800 page views on Associated Content\/Yahoo! Voices, and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for the first time.\u00a0 *** *** ~ G. Stolyarov II, July 17, 2014 ***…<\/p>\n

Read More Read More<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[292],"tags":[3113,3163,3162,3114,3149,40,3151,2656,3157,3121,3155,3161,3158],"class_list":["post-2755","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-actuarial-science","tag-continuous-distribution","tag-de-moivre-distribution","tag-exam-p","tag-expectation","tag-g-stolyarov-ii","tag-mathematical-expectation","tag-probability","tag-probability-density-function","tag-probability-theory","tag-random-variable","tag-uniform-distribution","tag-variance"],"_links":{"self":[{"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/posts\/2755","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/comments?post=2755"}],"version-history":[{"count":1,"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/posts\/2755\/revisions"}],"predecessor-version":[{"id":2756,"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/posts\/2755\/revisions\/2756"}],"wp:attachment":[{"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/media?parent=2755"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/categories?post=2755"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.rationalargumentator.com\/index\/wp-json\/wp\/v2\/tags?post=2755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}