{"id":7127,"date":"2011-11-01T02:26:08","date_gmt":"2011-11-01T02:26:08","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7127"},"modified":"2022-01-25T18:59:30","modified_gmt":"2022-01-25T18:59:30","slug":"a-custa-de-um-dado","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7127","title":{"rendered":"\u00c0 custa de um dado"},"content":{"rendered":"<p><ul id='GTTabs_ul_7127' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7127' class='GTTabs_curr'><a  id=\"7127_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7127' ><a  id=\"7127_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_7127'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>\u00c0 custa de um dado equilibrado defina uma experi\u00eancia a que associe uma vari\u00e1vel aleat\u00f3ria uniforme discreta.<\/p>\n<p>Determine a sua esperan\u00e7a matem\u00e1tica e a vari\u00e2ncia.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7127' onClick='GTTabs_show(1,7127)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7127'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/dado6.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"7128\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7128\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/dado6.jpg\" data-orig-size=\"215,234\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Dado\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/dado6.jpg\" class=\"alignright size-full wp-image-7128\" title=\"Dado\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/dado6.jpg\" alt=\"\" width=\"103\" height=\"112\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/dado6.jpg 215w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/dado6-137x150.jpg 137w\" sizes=\"auto, (max-width: 103px) 100vw, 103px\" \/><\/a>Consideremos a experi\u00eancia aleat\u00f3ria que consiste em lan\u00e7ar um dado c\u00fabico equilibrado e registar o n\u00famero de pontos inscrito na face voltada para cima.<\/p>\n<p>Consideremos a vari\u00e1vel aleat\u00f3ria uniforme discreta\u00a0$X$: &#8220;O valor da pontua\u00e7\u00e3o \u00e9 par&#8221;, tal que:<\/p>\n<ul>\n<li>$X=1$ quando o n\u00famero de pontos obtido \u00e9 um n\u00famero par;<\/li>\n<li>$X=0$ quando o n\u00famero de pontos obtido \u00e9 um n\u00famero \u00edmpar.<\/li>\n<\/ul>\n<p>A distribui\u00e7\u00e3o de probabilidades \u00e9 a seguinte:<\/p>\n<table class=\" aligncenter\" style=\"width: 30%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #1852e6 1px solid;\" align=\"middle\">$X={{x}_{i}}$<\/td>\n<td style=\"text-align: center; border: #1852e6 1px solid;\" align=\"middle\">0<\/td>\n<td style=\"text-align: center; border: #1852e6 1px solid;\" align=\"middle\">1<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #1852e6 1px solid;\" align=\"middle\">$P(X={{x}_{i}})$<\/td>\n<td style=\"text-align: center; border: #1852e6 1px solid;\" align=\"middle\">$\\frac{1}{2}$<\/td>\n<td style=\"text-align: center; border: #1852e6 1px solid;\" align=\"middle\">$\\frac{1}{2}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0A esperan\u00e7a matem\u00e1tica \u00e9\u00a0\\[E(X)=\\frac{1}{2}\\times 0+\\frac{1}{2}\\times 1=\\frac{1}{2}\\] e a vari\u00e2ncia \u00e9 \\[{{\\sigma }^{2}}=\\frac{1}{2}\\times {{(0-\\frac{1}{2})}^{2}}+\\frac{1}{2}\\times {{(1-\\frac{1}{2})}^{2}}=\\frac{1}{8}+\\frac{1}{8}=\\frac{1}{4}\\]<\/p>\n<\/p>\n<p><strong>Outra possibilidade<\/strong>:<\/p>\n<p>Consideremos a experi\u00eancia aleat\u00f3ria que consiste em lan\u00e7ar um dado c\u00fabico equilibrado e registar o n\u00famero de pontos inscrito na face voltada para cima.<\/p>\n<p>Consideremos a vari\u00e1vel aleat\u00f3ria uniforme discreta\u00a0$Y$: &#8220;N\u00famero de pontos inscrito na face voltada para cima&#8221;.<\/p>\n<p>A distribui\u00e7\u00e3o de probabilidades \u00e9 a seguinte:<\/p>\n<table class=\" aligncenter\" style=\"width: 70%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$Y={{y}_{i}}$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">1<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">2<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">3<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">4<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">5<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">6<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$P(Y={{y}_{i}})$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$\\frac{1}{6}$<\/td>\n<td style=\"text-align: center; border: #23db23 1px solid;\" align=\"middle\">$\\frac{1}{6}$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u00a0A esperan\u00e7a matem\u00e1tica \u00e9\u00a0\\[E(X)=\\frac{1}{6}\\times 1+\\frac{1}{6}\\times 2+\\frac{1}{6}\\times 3+\\frac{1}{6}\\times 4+\\frac{1}{6}\\times 5+\\frac{1}{6}\\times 6=\\frac{1}{6}\\times 21=\\frac{7}{2}\\] e a vari\u00e2ncia \u00e9 \\[{{\\sigma }^{2}}=\\frac{1}{6}\\times {{(1-\\frac{7}{2})}^{2}}+\\frac{1}{6}\\times {{(2-\\frac{7}{2})}^{2}}+\\frac{1}{6}\\times {{(3-\\frac{7}{2})}^{2}}+\\frac{1}{6}\\times {{(4-\\frac{7}{2})}^{2}}+\\frac{1}{6}\\times {{(5-\\frac{7}{2})}^{2}}+\\frac{1}{6}\\times {{(6-\\frac{7}{2})}^{2}}=\\frac{35}{12}\\]<\/p>\n<p>\u00ad<\/p>\n<blockquote>\n<p style=\"text-align: center;\">Diz-se que a vari\u00e1vel aleat\u00f3ria $X$, que toma um n\u00famero $n$ finito de valores ${{x}_{1}},\\ {{x}_{2}},\\ &#8230;,\\ {{x}_{n}}$, tem uma distribui\u00e7\u00e3o uniforme discreta se a probabilidade de qualquer desses valores for $p=P(X={{x}_{i}})=\\frac{1}{n}$.<\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7127' onClick='GTTabs_show(0,7127)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado \u00c0 custa de um dado equilibrado defina uma experi\u00eancia a que associe uma vari\u00e1vel aleat\u00f3ria uniforme discreta. Determine a sua esperan\u00e7a matem\u00e1tica e a vari\u00e2ncia. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":21018,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,227],"tags":[427,245,244,246],"series":[],"class_list":["post-7127","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","tag-variavel-aleatoria-uniforme-discreta"],"views":3205,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2011\/11\/12V1Pag105-35_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7127","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=7127"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7127\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21018"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7127"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7127"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7127"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}