{"id":286,"date":"2025-02-09T11:51:46","date_gmt":"2025-02-09T10:51:46","guid":{"rendered":"https:\/\/brunolovat.com\/?page_id=286"},"modified":"2025-02-10T19:13:08","modified_gmt":"2025-02-10T18:13:08","slug":"reponse-a-la-question-du-texte-sur-la-galette-des-rois","status":"publish","type":"page","link":"https:\/\/brunolovat.com\/index.php\/reponse-a-la-question-du-texte-sur-la-galette-des-rois\/","title":{"rendered":"R\u00e9ponse \u00e0 la question du texte sur la galette des rois"},"content":{"rendered":"\n

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


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

A-t-on vraiment :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\frac{\sigma(X)}{E(X)} < \frac{1}{\sqrt{8}} ?\]\"<\/p><\/p>\n\n\n\n

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

On peut expliciter \"E(X^2)\" en utilisant la valeur de \"E(X)\" que nous avons d\u00e9j\u00e0 trouv\u00e9e :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[E(X^2) = \sum_{i=k}^{n} i^2 p(X=i) \]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[= \sum_{i=k}^{n} (i+1) i p(X=i) - \sum_{i=k}^{n} i p(X=i)\]\"<\/p><\/p>\n\n\n\n

En ajustant l’expression de \"E(X)\", on trouve :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[ \frac{k}{k+1} \frac{k+1}{k+2} (n+2)(n+1) - E(X)\]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[E(X^2) = k (n+1) \frac{nk+n+k}{(k+2)(k+1)}\]\"<\/p><\/p>\n\n\n\n

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

Finalement la variance \"\sigma^2(X)\" de X a pour valeur :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\frac{k}{(k+2)(k+1)^2} (n+1) (n-k)\]\"<\/p><\/p>\n\n\n\n

et pour\"k \geq 2\", on a bien effectivement :<\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\frac{\sigma(X)}{E(X)} = \frac{1}{\sqrt{k(k+2)}} \sqrt{\frac{n-k}{n+1}} \]\"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\"\[\frac{\sigma(X)}{E(X)} < \frac{1}{\sqrt{k(k+2)}} \leq \frac{1}{\sqrt{8}}\]\"<\/p><\/p>\n\n\n\n

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

\"\end{document}\"<\/p>\n\n\n\n

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

A-t-on vraiment :     On peut expliciter en utilisant la valeur de que nous avons d\u00e9j\u00e0 trouv\u00e9e :         En ajustant l’expression de , on trouve :         Finalement la variance de X a pour valeur :     et pour, on a bien effectivement :     … <\/p>\n

Continuer la lecture de « R\u00e9ponse \u00e0 la question du texte sur la galette des rois »<\/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-286","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages\/286"}],"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=286"}],"version-history":[{"count":15,"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages\/286\/revisions"}],"predecessor-version":[{"id":326,"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/pages\/286\/revisions\/326"}],"wp:attachment":[{"href":"https:\/\/brunolovat.com\/index.php\/wp-json\/wp\/v2\/media?parent=286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}