{"id":265,"date":"2026-10-03T06:02:57","date_gmt":"2026-10-03T03:02:57","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/?p=265"},"modified":"2026-10-03T06:02:57","modified_gmt":"2026-10-03T03:02:57","slug":"calculateur-loi-exponentielle","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-exponentielle\/","title":{"rendered":"Calculateur de loi exponentielle"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Calculateur de distribution exponentielle<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"265\" data-favorite-title=\"Loi exponentielle\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-exponentielle\/\" data-favorite-excerpt=\"Calculez les probabilit\u00e9s de la loi exponentielle \u00e0 partir d\u2019un taux \u03bb ou d\u2019un temps d\u2019attente moyen : densit\u00e9, P(X \u2264 t), P(X &gt; t), probabilit\u00e9 sur un intervalle, moyenne et m\u00e9diane. Saisissez le taux, un temps et, \u00e9ventuellement, la borne sup\u00e9rieure d\u2019un intervalle.\" aria-label=\"Ajouter aux favoris\" data-favorite-icon=\"icon icon-wpcalc-distribution-histogram\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-exponentielle\/\" method=\"POST\" class=\"calculator\" id=\"calculator-265\" data-post=\"265\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">D\u00e9finir le param\u00e8tre par<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"dfinir_le_paramtre_par\"><option value=\"rate\" selected>Taux \u03bb (\u00e9v\u00e9nements par unit\u00e9 de temps)<\/option><option value=\"mean\">Temps d\u2019attente moyen \u03bc<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Taux (\u03bb)<\/label><div class=\"term\">\u03bb<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"taux_\" id=\"field-2\" aria-describedby=\"calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"2\"\/><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"par\" aria-label=\"par\"><option value=\"1\">par seconde<\/option><option value=\"60\">par minute<\/option><option value=\"3600\" selected>par heure<\/option><option value=\"86400\">par jour<\/option><option value=\"31536000\">par an<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-2\">Nombre moyen d\u2019\u00e9v\u00e9nements par unit\u00e9 de temps.<\/small><\/div><div class=\"field has-term\" id=\"input-3\" hidden><label for=\"field-4\">Temps d\u2019attente moyen (\u03bc)<\/label><div class=\"term\">\u03bc<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"temps_dattente_moyen_\" id=\"field-4\" aria-describedby=\"calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"30\"\/><div class=\"grid grid-cols-1\"><select id=\"field-5\" name=\"en\" aria-label=\"en\"><option value=\"1\">secondes<\/option><option value=\"60\" selected>minutes<\/option><option value=\"3600\">heures<\/option><option value=\"86400\">jours<\/option><option value=\"31536000\">ann\u00e9es<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-4\">Temps moyen entre deux \u00e9v\u00e9nements cons\u00e9cutifs (\u03bb = 1\/\u03bc).<\/small><\/div><div class=\"is-single field\" id=\"input-4\"><span class=\"font-semibold\">Temps<\/span><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-7\">Temps (t)<\/label><div class=\"term\">t<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"temps_t\" id=\"field-7\" aria-describedby=\"calc-info-7\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"1\"\/><div class=\"grid grid-cols-1\"><select id=\"field-8\" name=\"en\" aria-label=\"en\"><option value=\"1\">secondes<\/option><option value=\"60\">minutes<\/option><option value=\"3600\" selected>heures<\/option><option value=\"86400\">jours<\/option><option value=\"31536000\">ann\u00e9es<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-7\">Le temps d\u2019attente qui vous int\u00e9resse.<\/small><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-9\">Fin de l\u2019intervalle (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"fin_de_lintervalle_b\" id=\"field-9\" aria-describedby=\"calc-info-9\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"2\"\/><div class=\"grid grid-cols-1\"><select id=\"field-10\" name=\"en\" aria-label=\"en\"><option value=\"1\">secondes<\/option><option value=\"60\">minutes<\/option><option value=\"3600\" selected>heures<\/option><option value=\"86400\">jours<\/option><option value=\"31536000\">ann\u00e9es<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-9\">Facultatif : renseignez ce champ pour obtenir \u00e9galement P(t \u2264 X \u2264 b). Laissez-le vide sinon.<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalculer\" id=\"calculate-button\" data-post=\"265\">Calculer<\/button><button type=\"reset\">R\u00e9initialiser<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculer automatiquement<\/label><small>Si activ\u00e9, le r\u00e9sultat se met \u00e0 jour automatiquement d\u00e8s que vous modifiez une valeur.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> R\u00e9sultats de la distribution exponentielle<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Densit\u00e9 f(t)<\/span><span class=\"result-term\">f(t)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Probabilit\u00e9 cumul\u00e9e P(X \u2264 t)<\/span><span class=\"result-term\">F(t)<\/span><\/span><span class=\"field-value\" id=\"result-2\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Probabilit\u00e9 de survie P(X &gt; t)<\/span><span class=\"result-term\">S(t)<\/span><\/span><span class=\"field-value\" id=\"result-3\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Probabilit\u00e9 sur l\u2019intervalle P(t \u2264 X \u2264 b)<\/span><\/span><span class=\"field-value\" id=\"result-4\" data-unit=\"%\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Temps d\u2019attente moyen E(X)<\/span><span class=\"result-term\">E(X)<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-6\"><span class=\"field-title\"><span>Formule, m\u00e9diane et interpr\u00e9tation<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copier le lien vers ce calcul<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Laisser un commentaire<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"265\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Que pensez-vous de ce r\u00e9sultat\u00a0?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">J\u2019aime <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Utile <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confus <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">D\u00e9\u00e7u <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inexact <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Formules de la loi exponentielle<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formule<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ f(t) = \\lambda e^{-\\lambda t}, \\qquad F(t) = P(X \\le t) = 1 - e^{-\\lambda t} \\]\n\\[ S(t) = P(X &gt; t) = e^{-\\lambda t}, \\qquad P(a \\le X \\le b) = e^{-\\lambda a} - e^{-\\lambda b} \\]\n\\[ E(X) = \\frac{1}{\\lambda}, \\qquad \\text{m\u00e9diane} = \\frac{\\ln 2}{\\lambda} \\]<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>O\u00f9&nbsp;:<\/strong><\/p>\n<ul>\n<li>\\(\\lambda\\) \u2014 param\u00e8tre de taux : nombre moyen d\u2019\u00e9v\u00e9nements par unit\u00e9 de temps<\/li>\n<li>\\(t\\) \u2014 temps d\u2019attente consid\u00e9r\u00e9<\/li>\n<li>\\(f(t)\\) \u2014 densit\u00e9 de probabilit\u00e9 au temps \\(t\\)<\/li>\n<li>\\(F(t)\\) \u2014 probabilit\u00e9 cumul\u00e9e que l\u2019attente ne d\u00e9passe pas \\(t\\)<\/li>\n<li>\\(S(t)\\) \u2014 probabilit\u00e9 que l\u2019attente d\u00e9passe \\(t\\)<\/li>\n<\/ul>\n<p>La loi exponentielle mod\u00e9lise le temps entre des \u00e9v\u00e9nements cons\u00e9cutifs qui surviennent ind\u00e9pendamment \u00e0 un taux moyen constant. Son unique param\u00e8tre, \\(\\lambda\\), d\u00e9termine enti\u00e8rement la loi&nbsp;; le temps d\u2019attente moyen est simplement \\(1\/\\lambda\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Loi exponentielle \u2013 exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple&nbsp;:<\/strong> un centre d\u2019appels re\u00e7oit en moyenne 2 appels par heure. Quelle est la probabilit\u00e9 que le prochain appel arrive dans l\u2019heure, et quelle est la probabilit\u00e9 qu\u2019il arrive entre 1 et 2 heures ?<\/p>\n<p>\\[ \\lambda t = 2 \\times 1 = 2 \\]<\/p>\n<p>\\[ F(1) = 1 - e^{-2} = 1 - 0.1353 = 0.8647 \\]<\/p>\n<p>\\[ P(1 \\le X \\le 2) = e^{-2} - e^{-4} = 0.1353 - 0.0183 = 0.1170 \\]<\/p>\n<p><strong>R\u00e9sultat&nbsp;:<\/strong> P(X \u2264 1 h) = 86,47&nbsp;%, P(X &gt; 1 h) = 13,53&nbsp;% et P(1 h \u2264 X \u2264 2 h) = 11,70&nbsp;%. La densit\u00e9 \u00e0 t = 1 h est f(1) = 2e\u207b\u00b2 \u2248 0,2707 par heure, le temps d\u2019attente moyen est de 0,5 heure (30 minutes) et la m\u00e9diane est ln(2)\/2 \u2248 0,3466 heure (environ 20,8 minutes). Ce sont exactement les valeurs affich\u00e9es par le calculateur avec ses param\u00e8tres par d\u00e9faut.<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur<\/h2>\n<ol>\n<li>Ouvrez <strong>D\u00e9finir le param\u00e8tre par<\/strong> et choisissez de saisir soit le taux \\(\\lambda\\) (\u00e9v\u00e9nements par unit\u00e9 de temps), soit le temps d\u2019attente moyen \\(\\mu\\). Seul le champ correspondant s\u2019affiche.<\/li>\n<li>Si vous avez choisi le taux, renseignez <strong>Taux (\u03bb)<\/strong> et s\u00e9lectionnez l\u2019unit\u00e9 de temps correspondante (par seconde, par minute, par heure, par jour ou par an). Si vous avez choisi la moyenne, renseignez plut\u00f4t <strong>Temps d\u2019attente moyen (\u03bc)<\/strong> et son unit\u00e9 ; le calculateur effectue la conversion avec \\(\\lambda = 1\/\\mu\\).<\/li>\n<li>Saisissez le <strong>Temps (t)<\/strong> \u2014 le temps d\u2019attente qui vous int\u00e9resse \u2014 et s\u00e9lectionnez son unit\u00e9 sous \u00ab&nbsp;en&nbsp;\u00bb.<\/li>\n<li>Renseignez \u00e9ventuellement <strong>Borne sup\u00e9rieure de l\u2019intervalle (b)<\/strong> pour obtenir aussi la probabilit\u00e9 \\(P(t \\le X \\le b)\\). Laissez ce champ vide pour ignorer ce r\u00e9sultat.<\/li>\n<li>Consultez les r\u00e9sultats : la densit\u00e9 \\(f(t)\\), la probabilit\u00e9 cumul\u00e9e \\(P(X \\le t)\\), la probabilit\u00e9 que l\u2019attente d\u00e9passe \\(t\\), \\(P(X &gt; t)\\), la probabilit\u00e9 sur l\u2019intervalle, le temps d\u2019attente moyen et une ligne r\u00e9capitulative indiquant la m\u00e9diane, le 90e percentile du temps d\u2019attente et une interpr\u00e9tation en termes simples.<\/li>\n<\/ol>\n<h2>Qu\u2019est-ce que la loi exponentielle&nbsp;?<\/h2>\n<p>La loi exponentielle d\u00e9crit le temps d\u2019attente jusqu\u2019au prochain \u00e9v\u00e9nement dans un processus o\u00f9 les \u00e9v\u00e9nements surviennent ind\u00e9pendamment \u00e0 un taux moyen constant&nbsp;: un <em>processus de Poisson<\/em>. Elle s\u2019applique notamment au temps entre les arriv\u00e9es de clients, au temps avant la d\u00e9sint\u00e9gration d\u2019une particule radioactive, \u00e0 la dur\u00e9e de vie d\u2019un composant \u00e9lectronique ou au temps avant la prochaine panne d\u2019un serveur.<\/p>\n<p>Cette loi est d\u00e9finie par un seul param\u00e8tre, le taux \\(\\lambda\\). La densit\u00e9 de probabilit\u00e9 \\(f(t) = \\lambda e^{-\\lambda t}\\) est maximale en \\(t = 0\\) et d\u00e9cro\u00eet progressivement : les courtes attentes sont donc les plus probables. La fonction de r\u00e9partition \\(F(t) = 1 - e^{-\\lambda t}\\) donne la probabilit\u00e9 que l\u2019attente se termine au plus tard au temps \\(t\\) ; son compl\u00e9ment, la fonction de survie \\(S(t) = e^{-\\lambda t}\\), donne la probabilit\u00e9 d\u2019attendre encore apr\u00e8s \\(t\\).<\/p>\n<p>La moyenne et l\u2019\u00e9cart-type valent tous deux \\(1\/\\lambda\\), mais la m\u00e9diane \\(\\ln(2)\/\\lambda \\approx 0.693\/\\lambda\\) est plus petite. En pratique, cela signifie que plus de la moiti\u00e9 des attentes se terminent avant la dur\u00e9e moyenne. Une longue attente reste possible, mais devient de plus en plus rare : par exemple, seules 5 % des attentes d\u00e9passent \\(3\/\\lambda\\).<\/p>\n<p>Une propri\u00e9t\u00e9 caract\u00e9ristique de la loi exponentielle est l\u2019<strong>absence de m\u00e9moire<\/strong> : \\(P(X &gt; s + t \\mid X &gt; s) = P(X &gt; t)\\). Si la dur\u00e9e de vie d\u2019une ampoule suit une loi exponentielle et qu\u2019elle a d\u00e9j\u00e0 fonctionn\u00e9 pendant un an, ses chances de fonctionner encore un an sont exactement les m\u00eames que lorsqu\u2019elle \u00e9tait neuve. La loi exponentielle est ainsi l\u2019\u00e9quivalent continu de la loi g\u00e9om\u00e9trique et le pendant naturel de la <a href=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-de-poisson\/\">loi de Poisson<\/a>, qui d\u00e9crit le <em>nombre<\/em> d\u2019\u00e9v\u00e9nements sur une p\u00e9riode fixe plut\u00f4t que le temps entre ces \u00e9v\u00e9nements.<\/p>\n<h2>Hypoth\u00e8ses, limites et conseils<\/h2>\n<ul>\n<li><strong>Taux constant.<\/strong> Le mod\u00e8le suppose que les \u00e9v\u00e9nements surviennent \u00e0 tout moment \u00e0 un taux moyen constant. Si ce taux varie (par exemple, arriv\u00e9es aux heures de pointe ou usure de pi\u00e8ces m\u00e9caniques), la loi exponentielle sous-estimera ou surestimera les probabilit\u00e9s.<\/li>\n<li><strong>Ind\u00e9pendance.<\/strong> Un \u00e9v\u00e9nement ne doit pas influencer le moment o\u00f9 survient le suivant. Les regroupements d\u2019\u00e9v\u00e9nements ne respectent pas cette hypoth\u00e8se.<\/li>\n<li><strong>Absence de m\u00e9moire.<\/strong> Si le processus suit r\u00e9ellement une loi exponentielle, le temps d\u2019attente d\u00e9j\u00e0 \u00e9coul\u00e9 ne donne aucune information sur le temps restant. La dur\u00e9e de vie r\u00e9elle des machines et des \u00eatres humains ne pr\u00e9sente g\u00e9n\u00e9ralement pas cette propri\u00e9t\u00e9 ; des mod\u00e8les de fiabilit\u00e9 comme la loi de Weibull conviennent mieux dans ces cas.<\/li>\n<li><strong>Unit\u00e9s.<\/strong> Le calculateur convertit toutes les unit\u00e9s en interne&nbsp;: vous pouvez donc saisir un taux par heure et un temps en minutes sans faire de calcul vous-m\u00eame. Une ann\u00e9e est consid\u00e9r\u00e9e comme \u00e9gale \u00e0 365 jours.<\/li>\n<li><strong>Densit\u00e9 et probabilit\u00e9.<\/strong> \\(f(t)\\) est une densit\u00e9 par unit\u00e9 de temps, et non une probabilit\u00e9&nbsp;: elle peut d\u00e9passer 1. Les probabilit\u00e9s proprement dites sont \\(F(t)\\), \\(S(t)\\) et la probabilit\u00e9 sur un intervalle.<\/li>\n<li>Si vous souhaitez mod\u00e9liser un nombre d\u2019\u00e9v\u00e9nements plut\u00f4t qu\u2019un temps d\u2019attente, utilisez le <a href=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-de-poisson\/\">calculateur de loi de Poisson<\/a> ; pour des r\u00e9sultats oui\/non sur un nombre fixe d\u2019essais, consultez le <a href=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-binomiale\/\">calculateur de loi binomiale<\/a> et, pour des donn\u00e9es continues sym\u00e9triques, le <a href=\"https:\/\/wpcalc.com\/fr\/statistiques\/calculateur-loi-normale\/\">calculateur de loi normale<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculez les probabilit\u00e9s de la loi exponentielle \u00e0 partir d\u2019un taux \u03bb ou d\u2019un temps d\u2019attente moyen : densit\u00e9, P(X \u2264 t), P(X > t), probabilit\u00e9 sur un intervalle, moyenne et m\u00e9diane. Saisissez le taux, un temps et, \u00e9ventuellement, la borne sup\u00e9rieure d\u2019un intervalle.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[28],"tags":[56],"class_list":["post-265","post","type-post","status-publish","format-standard","hentry","category-statistiques","tag-distributions"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/265","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/comments?post=265"}],"version-history":[{"count":5,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/265\/revisions"}],"predecessor-version":[{"id":5063,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/265\/revisions\/5063"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}