{"id":197789,"date":"2025-03-07T16:22:46","date_gmt":"2025-03-07T16:22:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=197789"},"modified":"2025-03-07T16:22:48","modified_gmt":"2025-03-07T16:22:48","slug":"in-usual-notations-if-u5x20-and-u-100-and-variance-of-u-40-then-what-are-the-mean-and-variance-of-x","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/07\/in-usual-notations-if-u5x20-and-u-100-and-variance-of-u-40-then-what-are-the-mean-and-variance-of-x\/","title":{"rendered":"In usual notations If u=5x+20 and u = 100 and variance of u =40 then what are the mean and variance of x"},"content":{"rendered":"\n<p>In usual notations If u=5x+20 and u = 100 and variance of u =40 then what are the mean and variance of x ?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Given the equations for ( u ):<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>( u = 5x + 20 ) (Equation 1)<\/li>\n\n\n\n<li>( u = 100 ) (Equation 2)<\/li>\n\n\n\n<li>Variance of ( u = 40 )<\/li>\n<\/ol>\n\n\n\n<p>We are tasked with finding the mean and variance of ( x ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Find the value of ( x ) from ( u )<\/h3>\n\n\n\n<p>From Equation 2, we know that ( u = 100 ). Now substitute this into Equation 1:<\/p>\n\n\n\n<p>[<br>100 = 5x + 20<br>]<\/p>\n\n\n\n<p>Solving for ( x ):<\/p>\n\n\n\n<p>[<br>100 &#8211; 20 = 5x \\quad \\Rightarrow \\quad 80 = 5x \\quad \\Rightarrow \\quad x = \\frac{80}{5} = 16<br>]<\/p>\n\n\n\n<p>Thus, the mean of ( x ), denoted as ( \\mu_x ), is 16.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Find the variance of ( x )<\/h3>\n\n\n\n<p>The variance of a variable is related to how it scales under a linear transformation. In this case, the transformation from ( x ) to ( u ) is given by ( u = 5x + 20 ), which is a linear transformation with slope 5. We know that variance scales by the square of the coefficient when we transform a variable linearly.<\/p>\n\n\n\n<p>From the given information, the variance of ( u ) is 40. The variance of ( u ) is related to the variance of ( x ) by the equation:<\/p>\n\n\n\n<p>[<br>\\text{Var}(u) = 5^2 \\times \\text{Var}(x)<br>]<\/p>\n\n\n\n<p>Substitute the given variance of ( u ):<\/p>\n\n\n\n<p>[<br>40 = 25 \\times \\text{Var}(x)<br>]<\/p>\n\n\n\n<p>Solving for ( \\text{Var}(x) ):<\/p>\n\n\n\n<p>[<br>\\text{Var}(x) = \\frac{40}{25} = 1.6<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The mean of ( x ) is ( \\mu_x = 16 ).<\/li>\n\n\n\n<li>The variance of ( x ) is ( \\text{Var}(x) = 1.6 ).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The problem involves a linear relationship between ( x ) and ( u ), given by ( u = 5x + 20 ). The value of ( x ) is directly determined by substituting the given value of ( u ) into the equation. Since the variance of ( u ) is related to the variance of ( x ) by the square of the coefficient in the linear transformation, we use this relationship to find the variance of ( x ). The key steps involve solving for the value of ( x ) and then applying the property of variance under linear transformations to obtain the final result.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In usual notations If u=5x+20 and u = 100 and variance of u =40 then what are the mean and variance of x ? The correct answer and explanation is : Given the equations for ( u ): We are tasked with finding the mean and variance of ( x ). Step 1: Find the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-197789","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197789","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=197789"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/197789\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=197789"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=197789"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=197789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}