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    "\n",
    "## Exercice:\n",
    "\n",
    "Sans utiliser de boucles :\n",
    "\n",
    "  - Créer une matrice (5x6) aléatoire\n",
    "  - Remplacer une colonne sur deux (en partant de la première) par sa valeur moins le double de la colonne suivante\n",
    "  - Remplacer les valeurs négatives par 0 en utilisant un masque binaire\n",
    "  - Calculer l'approximation de $\\pi$ avec la formule de Wallis: \n",
    "     $$ \\pi = 2\\prod_{i=1}^{\\infty} \\frac{4i^2}{4i^2-1}$$ \n",
    "  - Examen 2017: approximation de ln(1+x) et visualisation de l'écart (cumsum, cumprod)\n",
    "      $$ ln(1+x) = \\sum_{i=1}^{\\infty} \\frac{(-1)^{i+1}}{i}x^i$$\n",
    "  - Examen 2018: approximation Monte Carlo de $\\pi$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.65064807  0.55031481  0.12193933  0.23659625  0.41359663  0.26513948]\n",
      " [-0.22053636 -0.08718648  0.62183885 -0.23879317 -0.2774651   0.01971086]\n",
      " [ 0.10584085 -0.09256091  0.57611705  0.66397284  0.36930593  0.02355765]\n",
      " [-0.18488314 -0.10033475  0.57341865  0.19488033  0.27488565 -0.21414899]\n",
      " [ 0.32949501  0.18343629  0.29262966 -0.08782539  0.44038484  0.51826797]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0.65064807 0.55031481 0.12193933 0.23659625 0.41359663 0.26513948]\n",
      " [0.         0.         0.62183885 0.         0.         0.01971086]\n",
      " [0.10584085 0.         0.57611705 0.66397284 0.36930593 0.02355765]\n",
      " [0.         0.         0.57341865 0.19488033 0.27488565 0.        ]\n",
      " [0.32949501 0.18343629 0.29262966 0.         0.44038484 0.51826797]]\n"
     ]
    }
   ],
   "source": [
    "\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.44998156  0.55031481 -0.35125317  0.23659625 -0.11668233  0.26513948]\n",
      " [-0.0461634  -0.08718648  1.09942519 -0.23879317 -0.31688683  0.01971086]\n",
      " [ 0.29096268 -0.09256091 -0.75182862  0.66397284  0.32219063  0.02355765]\n",
      " [ 0.01578636 -0.10033475  0.18365799  0.19488033  0.70318363 -0.21414899]\n",
      " [-0.03737757  0.18343629  0.46828044 -0.08782539 -0.59615109  0.51826797]]\n"
     ]
    }
   ],
   "source": [
    "\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.1415141186819566\n"
     ]
    }
   ],
   "source": [
    "# wallis\n",
    "\n",
    "print(res)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# approximation de pi par sampling disque/carré unité (Exam 2018)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "pi = "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14032"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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