Loi de grands nombres.ipynb

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     "source": [
      "# Loi de grands nombres\n",
      "\n",
      "Comme d'habitude, on a un [article sur Wikipedia](http://fr.wikipedia.org/wiki/Loi_des_grands_nombres) et une vid\u00e9o de l'Acad\u00e9mie Khan."
     ]
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      "from IPython.display import HTML, YouTubeVideo\n",
      "def IFrame(url, width=800, height=400):\n",
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      "    \n",
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        "        <iframe\n",
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        "            src=\"http://www.youtube.com/embed/Af6Lm9OV2Dc\"\n",
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      "$$\\begin{aligned}\n",
      "\\bar{X}_n &= \\frac{X_1 + X_2 + \\cdots + X_n}{n} \\\\\n",
      "\\bar{X}_n &\\rightarrow E(X) \\quad &\\text{quand }n\\rightarrow\\infty\n",
      "\\end{aligned}$$\n",
      "\n",
      "## La Loi faible des grands nombres\n",
      "\n",
      "La preuve math\u00e9matique de la loi faible est vraiment moins difficile que celle de la loi forte."
     ]
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      "$$\\forall \\varepsilon > 0, \\lim_{n\\rightarrow +\\infty}{P\\left(\\left|\\frac{X_1 + X_2 + \\cdots + X_n}{n} - E(X)\\right|\\geqslant \\varepsilon\\right)} = 0$$\n",
      "\n",
      "Pour prouver cette loi, il nous faudra l'[In\u00e9galit\u00e9 de Bienaym\u00e9-Techebychev](http://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Markov#Corollaire) :\n",
      "\n",
      "$$P(|Y-E(Y)| \\geqslant \\varepsilon) \\leqslant \\frac{V(Y)}{\\varepsilon^2}$$\n",
      "\n",
      "Et pour prouver \u00e7a, on aurait besoin de l'[In\u00e9galite de Markov](http://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Markov). Mais une chose \u00e0 la fois.\n",
      "\n",
      "On va introduire $\\bar{X}_n$ comme suit : \n",
      "\n",
      "$$\\begin{aligned}\n",
      "Y &= \\bar{X}_n \\\\\n",
      "\\bar{X}_n &= \\frac{X_1 + X_2 + \\cdots + X_n}{n} \\\\\n",
      "E(\\bar{X}_n) &= E(X) \\quad &\\text{(ci-dessous)} \\\\\n",
      "V(\\bar{X}_n) &= \\frac{V(X)}{n} \\quad &\\text{(dans le notebook \u00ab \u00c9cart type \u00bb)}\n",
      "\\end{aligned}$$\n",
      "\n",
      "Ainsi, on a :\n",
      "\n",
      "$$P\\left(\\left|\\frac{X_1 + X_2 + \\cdots + X_n}{n} - E(X)\\right|\\geqslant \\varepsilon\\right) \\leqslant \\frac{V(X)}{n\\varepsilon^2}$$\n",
      "\n",
      "Si on veut diviser $\\varepsilon$ par 2, il faut multiplier $n$ par 4 !\n",
      "\n",
      "Et plusieurs d\u00e9tails pour me convaincre :\n",
      "\n",
      "$$\\begin{aligned}\n",
      "E(\\bar{X}_n) &= E\\left[\\frac{X_1 + X_2 + \\cdots + X_n}{n}\\right] \\\\\n",
      "             &= \\frac{E(X_1) + E(X_2) + \\cdots + E(X_n)}{n} \\\\\n",
      "             &= \\frac{nE(X)}{n} \\\\\n",
      "             &= E(X)\n",
      "\\end{aligned}$$\n",
      "\n",
      "## L'In\u00e9galite de Markov\n",
      "\n",
      "Comme d'habitude, on a [un article sur Wikipedia](http://fr.wikipedia.org/wiki/In%C3%A9galit%C3%A9_de_Markov). Mail il utilise une notation qui n'est m'est pas famili\u00e8re. J'ai trouv\u00e9 plusieurs explications sur [la page anglaise](http://en.wikipedia.org/wiki/Markov%27s_inequality). Soit $Z$ une variable al\u00e9atoire positive ou nulle.\n",
      "\n",
      "$$P(Z \\geqslant a) \\leqslant \\frac{E(Z)}{a}$$\n",
      "\n",
      "On d\u00e9fine $1_X$ comme\n",
      "\n",
      "$$\\begin{aligned}\n",
      "1_X &= 1 \\quad &\\text{quand X se produit} \\\\\n",
      "1_X &= 0 \\quad &\\text{quand X ne se produit pas}\n",
      "\\end{aligned}$$\n",
      "\n",
      "On peut voir que\n",
      "\n",
      "$$\\begin{aligned}\n",
      "\\forall \\omega\\in\\Omega, Z(\\omega) &\\geqslant a1_{Z(\\omega) \\geqslant a} \\\\\n",
      "E(Z) &\\geqslant E(a1_{Z \\geqslant a}) \\\\\n",
      "     &\\geqslant aP(Z \\geqslant a) \\\\\n",
      "P(Z \\geqslant a) &\\leqslant \\frac{E(Z)}{a}\n",
      "\\end{aligned}$$\n",
      "\n",
      "## Un corollaire de l'in\u00e9galite de Markov\n",
      "\n",
      "Soit $phi$ une function positive ou nulle, et croissante, telle que si $x \\geq y$, $\\phi(x) \\geq \\phi(y)$. Soit $Z$ \u00e9gal \u00e0 $\\phi(Y)$, et $a$ \u00e9gal \u00e0 $\\phi(b)$. Donc\n",
      "\n",
      "$$P(\\phi(Y) \\geqslant \\phi(b)) \\leqslant \\frac{E(\\phi(Y))}{\\phi(b)}$$\n",
      "\n",
      "quand $\\phi(b) \\neq 0$. Mais $x \\geq y$ quand $\\phi(x) \\geq \\phi(y)$, et donc $P(\\phi(x) \\geq \\phi(y)) \\leqslant P(x \\geq y)$. Cela nous donne\n",
      "\n",
      "$$P(Y \\geqslant b) \\leqslant \\frac{E(\\phi(Y))}{\\phi(b)}$$\n",
      "\n",
      "## In\u00e9galit\u00e9 de Bienaym\u00e9-Techebychev\n",
      "\n",
      "Soit $Y = X-E(X)$, $b = \\varepsilon$ et $\\phi(x) = x^2$.\n",
      "\n",
      "$$\\begin{aligned}\n",
      "P(X-E(X) \\geqslant \\varepsilon) &\\leqslant \\frac{E\\left[(X-E(X))^2\\right]}{\\varepsilon^2} \\\\\n",
      "                                &\\leqslant \\frac{Var(X)}{\\varepsilon^2}\n",
      "\\end{aligned}$$\n",
      "\n",
      "## La Loi forte des grands nombres\n",
      "\n",
      "La voil\u00e0, mais elle est vraiment plus difficile \u00e0 prouver :"
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