{"id":359,"date":"2025-03-01T19:45:23","date_gmt":"2025-03-01T18:45:23","guid":{"rendered":"https:\/\/brunolovat.com\/?page_id=359"},"modified":"2025-03-02T18:34:18","modified_gmt":"2025-03-02T17:34:18","slug":"reponse-a-la-question-posee-dans-larticle-combien-avons-nous-de-sosies-sur-terre","status":"publish","type":"page","link":"https:\/\/brunolovat.com\/index.php\/reponse-a-la-question-posee-dans-larticle-combien-avons-nous-de-sosies-sur-terre\/","title":{"rendered":"R\u00e9ponse \u00e0 la question pos\u00e9e dans l’article : \u00ab\u00a0Combien avons-nous de sosies sur Terre ?"},"content":{"rendered":"\n

<\/p>\n\n\n\n

Mon coll\u00e8gue affirme que si on suppose : \"0<\sigma ^{2}(X)\leq E(X)\" alors :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[P(X=0)\leq \frac{1}{2}.\]\"<\/p><\/p>\n\n\n\n

<\/p>\n\n\n\n

Cela est d\u00e9j\u00e0 vrai dans le cas qu’on vient d’\u00e9tudier o\u00f9 \"\sigma ^{2}(X)=E(X)=1\" et \"P(X=0)=\frac{1}{e}\leq \frac{1}{2}\" et le reste pour n’importe quelle variable \"X\" mod\u00e9lisant le nombre des sosies de Mona Lisa\u2026 <\/p>\n\n\n\n

La variable \"X\" \u00e9tant positive non nulle, sa moyenne est non nulle et quitte \u00e0 consid\u00e9rer la variable \"\frac{X}{E(X)}\", on peut supposer : \"E(X)=1\" et donc \"E(X^{2})=\sigma ^{2}(X)+1\leq 2.\" <\/p>\n\n\n\n

Dans le calcul de \"E(X)\" la valeur 0 prise par la variable \"X\" ne compte pas :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[E(X)=E(X \cdot 1_{(X>0)})=1\]\"<\/p><\/p>\n\n\n\n

et d’apr\u00e8s l’in\u00e9galit\u00e9 de Cauchy-Schwarz :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[1\leq E(X^{2})E(1_{(X>0)}^{2})\]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[1\leq (\sigma ^{2}+1)P(X>0)\]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\leq 2P(X>0)\]\"<\/p><\/p>\n\n\n\n

<\/p>\n\n\n\n

ainsi :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[P(X>0)\geq \frac{1}{2}, \text{ et } P(X=0)\leq \frac{1}{2}.\]\"<\/p><\/p>\n\n\n\n

Mon coll\u00e8gue a raison, la probabilit\u00e9 que Mona Lisa n’ait jamais eu de sosie est bien inf\u00e9rieure \u00e0 \"1/2\"\u2026<\/p>\n\n\n\n

Retour \u00e0 l’article.<\/a><\/strong><\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Mon coll\u00e8gue affirme que si on suppose : alors :     Cela est d\u00e9j\u00e0 vrai dans le cas qu’on vient d’\u00e9tudier o\u00f9 et et le reste pour n’importe quelle variable mod\u00e9lisant le nombre des sosies de Mona Lisa\u2026 La variable \u00e9tant positive non nulle, sa moyenne est non nulle et quitte \u00e0 consid\u00e9rer la … <\/p>\n

Continuer la lecture de « R\u00e9ponse \u00e0 la question pos\u00e9e dans l’article : \u00ab\u00a0Combien avons-nous de sosies sur Terre ? »<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-359","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages\/359"}],"collection":[{"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/comments?post=359"}],"version-history":[{"count":20,"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages\/359\/revisions"}],"predecessor-version":[{"id":389,"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages\/359\/revisions\/389"}],"wp:attachment":[{"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/media?parent=359"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}