{"id":2146,"date":"2020-03-13T10:11:25","date_gmt":"2020-03-13T09:11:25","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=2146"},"modified":"2020-10-02T18:34:57","modified_gmt":"2020-10-02T16:34:57","slug":"distance-moyenne-integrale-et-loi-continue","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/13\/distance-moyenne-integrale-et-loi-continue\/","title":{"rendered":"Distance moyenne, int\u00e9grale et loi continue"},"content":{"rendered":"\n<p>Voici un probl\u00e8me que les \u00e9l\u00e8ves de Terminale pourront comprendre&#8230; si la notion de loi uniforme leur dit quelque chose&#8230;<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_83 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/13\/distance-moyenne-integrale-et-loi-continue\/#Le_probleme\" >Le probl\u00e8me<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/13\/distance-moyenne-integrale-et-loi-continue\/#Comment_commencer\" >Comment commencer ?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/13\/distance-moyenne-integrale-et-loi-continue\/#Une_solution\" >Une solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2020\/03\/13\/distance-moyenne-integrale-et-loi-continue\/#Remarque_sur_la_variable_alpha\" >Remarque sur la variable \\(\\alpha\\)<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Le_probleme\"><\/span>Le probl\u00e8me<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Soient deux points A et B sur un cercle de rayon 1. Quelle est la distance moyenne entre ces deux points ?<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Comment_commencer\"><\/span>Comment commencer ?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Pour vous aider, voici une petite aide: fixons le point A et consid\u00e9rons le diam\u00e8tre du cercle d&#8217;extr\u00e9mit\u00e9 A. Notons O le centre du cercle. Alors, \\(\\widehat{AOB}\\) est une variable al\u00e9atoire continue prenant ses valeurs sur \\([0;\\pi]\\)&#8230; et la distance AB s&#8217;exprime comme une fonction de cette variable.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Une_solution\"><\/span>Une solution<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/loi-uniforme.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"467\" height=\"467\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/loi-uniforme.png\" alt=\"\" class=\"wp-image-2147\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/loi-uniforme.png 467w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/loi-uniforme-300x300.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/loi-uniforme-100x100.png 100w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2020\/03\/loi-uniforme-150x150.png 150w\" sizes=\"auto, (max-width: 467px) 100vw, 467px\" \/><\/a><\/figure><\/div>\n\n\n\n<p>D&#8217;apr\u00e8s le th\u00e9or\u00e8me d&#8217;Al-Kashi,$$AB^2=OA^2+OB^2-2\\times OA\\times OB\\times\\cos\\alpha$$c&#8217;est-\u00e0-dire ici (car OA = OB = 1):$$AB^2=2-2\\cos\\alpha.$$Ainsi,$$AB=\\sqrt{2-2\\cos\\alpha}.$$Or,$$\\cos(2x)=1-2\\sin^2x$$donc:$$\\sin x = \\sqrt{\\frac{1-\\cos(2x)}{2}}.$$En prenant \\(x=\\frac{\\alpha}{2}\\), on obtient:$$\\sin\\frac{\\alpha}{2}=\\sqrt{\\frac{1-\\cos\\alpha}{2}}= \\sqrt{\\frac{2-2\\cos\\alpha}{4}} .$$On d\u00e9duit alors que:$$AB=2\\sin\\frac{\\alpha}{2}.$$<\/p>\n\n\n\n<p>Le probl\u00e8me \u00e9tant sym\u00e9trique par rapport au diam\u00e8tre passant par A, on peut consid\u00e9rer en effet que l&#8217;angle \\(\\alpha\\) est compris entre 0 et \\(\\pi\\). Si on d\u00e9signe la distance AB par :$$f(\\alpha)=2\\sin\\frac{\\alpha}{2}$$alors la distance moyenne est la valeur moyenne de la fonction <em>f<\/em>:$$\\mu=\\frac{1}{\\pi-0}\\int_0^{\\pi}f(\\alpha)\\text{d}\\alpha=\\frac{2}{\\pi}\\int_0^{\\pi}\\sin\\frac{\\alpha}{2}\\text{d}\\alpha.$$On a alors:$$\\mu=\\frac{2}{\\pi}\\left[-2\\cos\\frac{\\alpha}{2}\\right]_0^\\pi=\\frac{4}{\\pi}.$$<\/p>\n\n\n\n<p>La distance moyenne est donc \u00e9gale \u00e0 \\(\\displaystyle\\frac{4}{\\pi}\\), c&#8217;est-\u00e0-dire environ 1,27.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Remarque_sur_la_variable_alpha\"><\/span>Remarque sur la variable \\(\\alpha\\)<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Nous avons vu que, dans la mesure o\u00f9 B est choisi au hasard, l&#8217;angle \\(\\alpha\\) est al\u00e9atoire. On peut consid\u00e9rer qu&#8217;elle suit la loi uniforme sur \\([0;\\pi]\\). Cela dit, cela n&#8217;a aucune importance pour notre probl\u00e8me. <\/p>\n\n\n\n<p>Mais si on se demande quelle est la probabilit\u00e9 que la longueur AB soit plus grande que 1, cela peut nous aider. En effet, $$AB&gt;1 \\iff \\alpha&gt;\\frac{\\pi}{3}.$$Si \\(\\alpha=\\frac{\\pi}{3}\\), le triangle AOB est \u00e9quilat\u00e9ral et donc AB=1.<\/p>\n\n\n\n<p>Ainsi,$$P(AB&gt;1)=P\\left(\\frac{\\pi}{3}&lt;\\alpha&lt;\\pi\\right)=\\frac{\\pi-\\frac{\\pi}{3}}{\\pi-0}=\\frac{2}{3}.$$En effet, si X suit la loi uniforme sur [<em>a<\/em> ; <em>b<\/em>] alors:$$P(c &lt; X &lt; d)=\\frac{d-c}{b-a}.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Voici un probl\u00e8me que les \u00e9l\u00e8ves de Terminale pourront comprendre&#8230; si la notion de loi uniforme leur dit quelque chose&#8230;<\/p>\n","protected":false},"author":1,"featured_media":3701,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,6],"tags":[60,191,104,192,190],"class_list":["post-2146","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-enseignement","category-mathematiques","tag-cercle","tag-distance","tag-integrales","tag-loi-continue","tag-moyenne"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - 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