{"id":7137,"date":"2011-11-08T16:10:48","date_gmt":"2011-11-08T16:10:48","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7137"},"modified":"2021-12-28T18:59:47","modified_gmt":"2021-12-28T18:59:47","slug":"a-lei-de-probabilidade-de-uma-variavel-aleatoria","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7137","title":{"rendered":"A lei de probabilidade de uma vari\u00e1vel aleat\u00f3ria"},"content":{"rendered":"<p><ul id='GTTabs_ul_7137' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7137' class='GTTabs_curr'><a  id=\"7137_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7137' ><a  id=\"7137_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7137'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>A lei de probabilidade de uma vari\u00e1vel aleat\u00f3ria $X$ \u00e9:<\/p>\n<table class=\" aligncenter\" style=\"width: 60%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">${{x}_{i}}$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">1<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">2<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">3<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">4<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">5<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">$P(X={{x}_{i}})$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">0,1<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">0,2<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">0,1<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">0,3<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">0,1<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">0,2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol>\n<li>Calcule a esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o de $X$.<\/li>\n<li>Uma vari\u00e1vel aleat\u00f3ria $Y$ toma os valores 3, 4, 5 e 6.\n<p>a) Qual \u00e9 a lei de probabilidade de $Y$, sabendo que:<br \/>\n$P(Y&gt;5)=0,5$; $P(Y&lt;5)=\\frac{1}{3}$ e $P(Y=3)=P(Y=4)$.<\/p>\n<p>b) Qual \u00e9 a esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o de $Y$?<\/p>\n<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7137' onClick='GTTabs_show(1,7137)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7137'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>A esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o de $X$ s\u00e3o, respetivamente:<br \/>\n\\[\\mu =0,1\\times 1+0,2\\times 2+0,1\\times 3+0,3\\times 4+0,1\\times 5+0,2\\times 6=3,7\\]<br \/>\n\\[\\sigma =\\sqrt{0,1\\times {{(1-3,7)}^{2}}+0,2\\times {{(2-3,7)}^{2}}+&#8230;+0,1\\times {{(5-3,7)}^{2}}+0,1\\times {{(6-3,7)}^{2}}}=\\sqrt{2,61}\\approx 1,62\\]<\/li>\n<li>a)<br \/>\nComo $P(Y&gt;5)=0,5$, ent\u00e3o $P(Y=6)=\\frac{1}{2}$.<\/p>\n<p>Se $P(Y&lt;5)=\\frac{1}{3}$, ent\u00e3o $P(Y=3)+P(Y=4)=\\frac{1}{3}$.<\/p>\n<p>Dado que, ainda, $P(Y=3)=P(Y=4)$, resulta $P(Y=3)=P(Y=4)=\\frac{1}{6}$.<\/p>\n<p>Finalmente, $P(Y=5)=1-P(Y=3)-P(Y=4)-P(Y=6)=\\frac{1}{6}$.<\/p>\n<p>Logo, a distribui\u00e7\u00e3o de probabilidades da vari\u00e1vel $Y$ \u00e9:<\/p>\n<table class=\" aligncenter\" style=\"width: 60%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">${{y}_{i}}$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">3<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">4<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">5<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">$P(Y={{y}_{i}})$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #d2691e 1px solid;\">$\\frac{1}{2}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>b)<br \/>\nA esperan\u00e7a matem\u00e1tica e o desvio padr\u00e3o da vari\u00e1vel aleat\u00f3ria $Y$ s\u00e3o:<br \/>\n\\[\\mu =\\frac{1}{6}\\times 3+\\frac{1}{6}\\times 4+\\frac{1}{6}\\times 5+\\frac{1}{2}\\times 6=5\\]<br \/>\n\\[\\sigma =\\sqrt{\\frac{1}{6}\\times {{(3-5)}^{2}}+\\frac{1}{6}\\times {{(4-5)}^{2}}+\\frac{1}{6}\\times {{(5-5)}^{2}}+\\frac{1}{2}\\times {{(6-5)}^{2}}}=\\sqrt{\\frac{4}{3}}=\\frac{2\\sqrt{3}}{3}\\approx 1,15\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7137' onClick='GTTabs_show(0,7137)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A lei de probabilidade de uma vari\u00e1vel aleat\u00f3ria $X$ \u00e9: ${{x}_{i}}$ 1 2 3 4 5 6 $P(X={{x}_{i}})$ 0,1 0,2 0,1 0,3 0,1 0,2 Calcule a esperan\u00e7a matem\u00e1tica e o desvio&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19177,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,245,244],"series":[],"class_list":["post-7137","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-probabilidades-e-combinatoria","tag-12-o-ano","tag-distribuicao-de-probabilidades","tag-variavel-aleatoria"],"views":1882,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat68.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7137","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7137"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7137\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7137"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}