4.Sorting in linear time.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "257ca062",
   "metadata": {},
   "source": [
    "### CDF/PDF for density\n",
    "* $f(a)$ is a PDF:\n",
    "$$Pr(a\\leq X \\leq b) = \\int_a^b f(x) dx$$\n",
    "* $F(a)$ is a CDF:\n",
    "$$F(a) = Pr(X\\leq a) = \\int_{-\\infty}^a f(x) dx$$\n",
    "* \n",
    "$$f(a)  = \\left. \\frac{d}{dx} F(x) \\right|_{x=a} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a7c34e1",
   "metadata": {},
   "source": [
    "### CDF for empirical distribution\n",
    "We have a sample $x_1,\\ldots,x_n$\n",
    "$$\\hat{F}(a) = \\frac{1}{n} |\\{i \\mbox{ such that } x_i \\leq a\\}|$$  \n",
    "* What about the PDF for empirical distribution? We cannot take the derivative...  The closest we have to and empirical pdf is the histogram."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "198151f0",
   "metadata": {},
   "source": [
    "### Calculating the probability of a segment\n",
    "* The true probability\n",
    "$$P(a\\leq X \\leq b) = \\int_a^b f(x) dx = \\int_{-\\infty}^b f(x)dx - \\int_{-\\infty}^a f(x)dx = $$  \n",
    "  \n",
    "* The empirical probability. We have a sample $x_1,\\ldots,x_n$\n",
    "$$\\hat{P}(a \\leq X \\leq b) = \\frac{1}{n} |\\{i \\mbox{ such that } a \\leq x_i \\leq b\\} = $$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "73f186b5",
   "metadata": {},
   "source": [
    "### Empirical CDFs vs. histograms\n",
    "* The histogram coverges to the density function\n",
    "* The emprirical CDF converges to the true CDF\n",
    "* The convergence of the CDF is much faster."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "4308dac8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "564cb554",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import norm\n",
    "def plot_normal(mu=1,sigma=1, n=20, m=100, plot_pdf_model=True,plot_cdf_model=True,\n",
    "               plot_pdf_empir=True,plot_cdf_empir=True):\n",
    "    s = np.random.normal(mu, sigma, n)\n",
    "\n",
    "    xmin=-4; xmax=6; delta=1/n\n",
    "    x=arange(xmin,xmax,delta)\n",
    "    if plot_cdf_model:\n",
    "        _cdf=norm.cdf(x,loc=mu,scale=sigma)\n",
    "        plot(x,_cdf)\n",
    "    if plot_pdf_model:\n",
    "        _pdf=norm.pdf(x,loc=mu,scale=sigma)\n",
    "        plot(x,_pdf)\n",
    "    grid()\n",
    "    if plot_cdf_empir:\n",
    "        q=sorted(s)\n",
    "        P=arange(0,1,1/s.shape[0])\n",
    "        plot(q,P)\n",
    "    if plot_pdf_empir:\n",
    "        plt.hist(s, 30, density=True);\n",
    "    \n",
    "    return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a5b35051",
   "metadata": {},
   "outputs": [
    {
     "data": {
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0oCt1CH/+8FHE4edqvQi0inJ6kWilutHU2sSSyudJtMZz3rhpdsdRqlta0JXqxuNrFmI5Gjg/71t2R1HqkPSQi1LdePqLp7G8A/nxCWfbHUWpQ9KCrlQXNuzZSHVgC6MTZ5Kdmmh3HKUOSQu6Ul24b+UzGOPgyslfszuKUj2iBV2pTrRarXy4exGO5gnMPkovAq1igxZ0pTrx5pb3aaWOEweeg9upvyYqNug7ValOPLJmAZY/lR9Pn213FKV6TAu6Uh1UNleyqX456f5pTMrrZ3ccpXpMC7pSHTz4yXyQAF8be7HdUZT6SrSgK9XBa1teI9A0kiun6ZmhKrZoQVeqnY1VX1IbKGZU8olkp+pCXCq2aEFXqp37Vy3AGAdXTJpjdxSlvjIt6Eq1McawpOR9HC2jueDoI+2Oo9RXpgVdqTbLStbQRCmT+52Kx6W/Gir26LtWqTb3rXwGYzm5bto37I6iVK9oQVeK4Lrnn1a/TXLrFKYNH2J3HKV6RQu6UsCC9W9gSQuzhl+IiNgdR6le0YKuFLBgwytYren8+PiZdkdRqte0oKs+r6Kxkp0tnzDINZ1BGcl2x1Gq17Sgqz7vH8sex2Dx7fGX2h1FqcOiBV31ad6Alzd3voC0jOWyKVPtjqPUYdGCrvq0l758DR91nND/GyS6nXbHUeqwaEFXfZYxhgc+fYRASw5zjzvL7jhKHTYt6KrPWl2+mgrvdjJaZzB1eJbdcZQ6bFrQVZ/1n3XPYgIJfHP8BTr3XMUFLeiqT2r2N/N+8Tv464/mkqmj7Y6jVEhoQVd90rvb36XVtDAutYAhmUl2x1EqJLSgqz7pP+texGrN5LvHzrA7ilIhowVd9TmVzZWsq16BNEzhnKN0IS4VP3pU0EVklohsFJEiEfl1N+2OE5GAiOjVdVXU+u+WNzFYnDz4TJI9LrvjKBUyhyzoIuIE7gXOASYAl4nIhC7a3Q4sCnVIpUJpwYbXsLzZfC//eLujKBVSPRmeTAOKjDFbAERkPnABsL5Du+uB54HjQppQqRAqbShlW8NnJLfOYtrI/vt3VBZByWrYsxl8DeBOgvRcyJ0KOUeBQ49OqujXk4KeC+xst10MTG/fQERygYuA0+mmoIvIXGAuQE5ODoWFhV8xblBDQ0OvHxurtM+h8eIXz0CSYXLjUN699x8MbVlObssqUgMV+9oEHB4clh/BAsDnzmD3oBmUDDmXlqSckObpSH/OfUO4+tyTgt7ZGRemw/YdwE3GmEB3J2gYY+YB8wDy8/NNQUFBz1J2UFhYSG8fG6u0z6Fxy8rfYclAbl52F8PHVeD0GBrLPezekUFThYdRSzbgdLoh4Ie6XbBjKZ4NCxm26VWG7XoN8r8PM26GpMyQ5tpLf859Q7j63JOCXgwMbbedB5R0aJMPzG8r5tnAuSLiN8a8FIqQSoVCdXMtVdk1fLOqlVFHl9NQkkDFZ2m0VHv2N3K62766IGt48Db5m1C7C96/HVY8BBtegTl3w1i9GIaKLj0p6CuAsSIyEtgFXApc3r6BMWbk3u9F5DHgVS3mKto8/catIIazG2opXpxF/a5EOv4BuuHI8d0+x/h33oaXfgRPfh1O+zWcdpMeX1dR45DvRGOMH7iO4OyVDcACY8w6EblGRK4Jd0ClDpsx8MFf+Wz3y2S3QtqL6dTvSqLzo4mHkDsF5r4Pky+H9/8Cz30P/N5QJ1aqV3o0CdcY8zrweof77u+i7fcOP5ZSIWJZsOhmqlc8wJJheQwvGoE0F/X66faP4A39xqWTw8s0vPMGxR9mYQIOxn+xITS5leoF/VtRxS9jYNHNsOw+Hs07DUvgopV7QvTkQtXGVEqWZZIyyMvQU6sQZ8e5AkpFlhZ0Fb/evx2W3Yc17Yc8YVpIsPKYvq06pC9RuzWZkqWZJA/0kXtCdXB2jFI20YKu4tOKh6Hwz3DMt1iQO5uAaxczcy/qzVHzQ6rbnkzZ6nTS8lrg9Z8H/zJQygZa0FX82foB/PdXMPYsOP8uHvtsAcby8LMTvxm2l6z+MpXK9amw6jFYcnfYXkep7mhBV3Fjw5HjKZo6Fv+8OXirYeOf17I4fyrFvo/J2TWYPWFeu6VibRqMnwNv3wKb3w3raynVGS3oKm6I0yLvlCoE2Lm4H5bfwRMnDkOcPi5c3hCJBHDhfZA9Dp77PlRvi8BrKrWfFnQVN3Km1JGQ4WfXkixaG1wYYMWEZlyN/ThjU1lkQiSkwqVPgrFgwXd1jrqKKC3oKj589hxZo5vYsz6VxrJEAD4YmUsgtYwpG5PC8mFol/qPhgv+BaVr4J1bI/nKqo/Tgq5iX9VWeOVGmircVHyetu/u56ZnYoyDy5d2XHooAsbPhvyr4ON74Mu3I//6qk/Sy7Wo2Bbww/NXgcPBro+zwATH4pWJyezO28WAsgEMqd0VsTjt14IRp2HEmS5cD3+DLW8MINDi1DNJVVjpCF3Fto/ugF2rYPYd+Jv2j08eO3EE4m7gwuVNtkUzAaHk4ywcbovB+bUcvOq0UqGlBV3FrrJ1UPgXmHgRHPW1fXdbCCsn1uNqymDmutCeGfpVeWvdVKwNnnSUPrzZ1iwq/mlBV7Ep0BpcxjYpE879+wG7XpswDCu1lOM/T46KN3jVphSaKt0MmlIL9bvtjqPiWDS835X66j78Z3AWyXn/gJT+B+x6ZVoCBDxc8ZENH4Z2xgily7IQl4FXf6pLA6iw0Q9FVczY+4FjQkYrI8+uoG5nEiVfuwm4aV+bL7MzqM7ZzohtQ8ho2WpT0oP56l1UrE0nx/k6fPYsTLrE7kgqDukIXcUWMQyeVkPA56BsVcZBux8+ZTDiCHDF4kobwnWvalMK5E2D/94EjaFaxlep/bSgq5iSOaqJpP6tlH2STsB34Nu3zuNm8+jdpFcO5qiSepsSdsMInH8neOvgrf+xO42KQ1rQVcxwJgQYOLmOxjIPdduTDtr/QMHQ4FTFj302pOuhnAlw4g3w6ZOwdbHdaVSc0YKuYkbOsXWI07B7ZSYdrwfa4hJWHl2Jp3YI534eoXVbeuvUX0LWCHj1Rl3rRYWUFnQVG7YUkjGimT0bUvHVH/xZ/lPTB2M8TZy9LCH639SeZDjv77CnKDhbR6kQifr3vlK0tsCrP8NX72TP+rROm7x3bACas/nm6i0RDtdLY2bCURfD4r9D5Zd2p1FxQgu6in4f3QFVm9m9KgNjHbxuYuERWXjT9jD5i0w8JhD5fL119m3gTtK56SpktKCr6LZnc3AUe9TXadyd2GmTp05NxvhTuPqDHREOd5jScmDm/8K2xbBmvt1pVBzQgq6ilzHw2s/AlRgczXbik2Gp1AwoY2TRMHKa7FuIq6c2HDn+wNu3b6ep0o1//o/ZNGkcLevW2R1RxTAt6CpqDSxfDFsK4YzfQdqgTts8dloWJuDhmveKIxsuZITdKzJxeiwGTq6zO4yKcVrQVXRqrmFM0cMw5FjI/36nTXZkJVCSW8bA4uGMqrF3VcXD4a11U7UxhczRTWS2brM7jophupaLihrtLw6RM7WGrLHNbH10Jy3/OKrT9nefnQ1mD1d8UBWpiGFT8Xka6cNamOh8Cfx/ApfH7kgqBukIXUWdxH4+ssY0sT3pBFqqOy9spRkutg+vIKtkDNOKo/xEoh4wAQe7V2WQFiiHpffaHUfFKC3oKrqIYfBxNfibHXyZPLPLZg+cnoMR+NbiKFyzpZcaShLZ7ZkAhbdD9Ta746gYpAVdRZWssY0kZvkpW51BwNH5NMXaRCfrx1aTXDmaU7dtj3DC8PoidTY4nPD6L3VuuvrKtKCrqOFKCjDg6HoaShKoL+68mAM8cFoeOH1cvKSZg08zim0tzkyYcTN8+SZsWGh3HBVjtKCrqJEzpRYRw+5VGXRcfGuveo+TlUdX46nJZfb6bRHNFzHTfgiDjg6um+6Nn0NKKvx6VNBFZJaIbBSRIhH5dSf7vyUia9tuS0Rkcuijqri2aRHpQ1uoXJ9Ga2PXk6/unZkH7ia+sdjE3eh8H6cLZt8RvP7ou3+yO42KIYcs6CLiBO4FzgEmAJeJyIQOzbYCpxljJgF/AOaFOqiKY74meP0XeGtd7Pkitctm5WlOVh9VTuKeEcz5PMZO8/+q8vKD8++XPwAln9qdRsWInozQpwFFxpgtxhgfMB+4oH0DY8wSY8zeMzuWAnmhjani2gd/hZodlK7MgE4W39rr3pmDMA4/33knEL+j8/bO+B0kZwfXTbdiaNExZRsxh/gkXUQuBmYZY65u2/4OMN0Yc10X7X8BHLm3fYd9c4G5ADk5OVPnz+/dgkQNDQ2kpnY9kotH8drn5MYd5K+8kbKc01gTOHCaom/gQDzl5QBsYw/3pjxPYv0kbpXjkTit6L6BA0kfMGDf9sCyD5iw4e98OWYuu/LOszFZ+MTre7s7h9PnGTNmrDLG5He2rydninb2q9PpvwIiMgO4Cji5s/3GmHm0HY7Jz883BQUFPXj5gxUWFtLbx8aquOyzZcHjsyExncHfeZCaqaccsHv79dcx/O57ALj9qmysxGRueKGcEaX32JE2IrZffx0F3/jG/jvMafDv1Yzd8TRj5/wM0gfbFy5M4vK9fQjh6nNPCnoxMLTddh5Q0rGRiEwCHgLOMcboJc3Voa16FLZ/BHPugZTsLpstHZ1E5cAacjdNJb90WQQD2qP9EggA7lQ/o86pp+HGY9i1pB/jv9hgUzIV7XpyDH0FMFZERoqIB7gUOGCCrIgMA14AvmOM2RT6mCru1BbDW7fAqAI49ttdNvO64F+z3Vi+LH7y9ubI5YsirQ0u9qwPrvWSMqjF7jgqih1yhG6M8YvIdcAiwAk8YoxZJyLXtO2/H/gd0B/4lwQPbvq7Osaj+q79I0/D0FOrSB7oY8ud62m9reOkqf2em55KS3ITk5ZPYlTth5EJGoX2bEglfXgzg/JrobU5eKUjpTro0WqLxpjXgdc73Hd/u++vBg76EFSpzqQPbyZ1iJfdq9K7nXNeK428cqIfq3401324JoIJo4+xhN0rMxh++h54709w1h/tjqSikJ4pqiLKmRAgZ0otTRVuqotSum37jOMTAk7DuR9m0U/PmKSpPIHqomRYcg/sWGp3HBWFtKCrCDIMmlqLw2UoXZEJpuu5h5sHOvkycR0JZZP59toVkYsY5co/TYfMofDSj8DXaHccFWW0oKuISR/eTPqwFio/T8NX5+627V3nZGOsBK55ew8ePalmH8vvgAv+BVVb4O3f2x1HRRm9YpGKjJqdDJpaS1OFp9vT+wE+GJdC6ZA99K+ewck734pQwNix4Zy55BybQj/msf3252kqSzhgv05r7Lt0hK7Cz7KChwgESpZmdnuopdkN885OwHizuFKOjFzGGFO+Ng1vnZMh02pwuC2746gooQVdhd/H98C2xZStzuh2VgvAP84ZhC+ljoJlOQxy6QUeumICDkqWZeFKCjAov4YuTt5WfYwWdBVepWvh3T/AkbOp3dr93Ol3jkpizYQ9JJeN45qP1kUoYOxq2eOh4vM0Moa3kDGqye44KgpoQVfh01ILz14RXDHw/Lvo6qIVAE0eeHAWWM253PxiOU4dcfbIng2pNO72MGhKHZ70VrvjKJtpQVfhYQwsvB6qt8M3HoWU/t02v3vmECx3K6d/PJAjqisiFDIOGKFkaRaWX8g9sRpx6j+EfZkWdBUey+fB+pdh5i0w7Phumy4Zlc7KoytJqhjND1esjFDA+OFvcVKyNJPETD+DptTqxaX7MC3oKvSKV8Ki38IR58AJ13fbtMXp4J7ZieBP5Zbny3AanbHRG427E6lcl0rm6CZY+bDdcZRNtKCr0KrdBfMvh/QhcOG/wNH9W+yP543Fn1LF7MWZjKquiUzGOFXxeRr1uxKCF5fevsTuOMoGWtBV6PiaYP5lwVPSL38Gkvt12/y5SaPYOH47/UuHcMXyLREKGcfajqeTNQIWfDe4RLHqU7Sgq9AwJnjyUOla+PrDMHB8t83XDcjmmZkNOFsTuPW5g66XonrJanXApU9Baws8fSm01NkdSUWQnvqvQuOd/4X1L1H2aTpV838K/LTLpmWpCfzhMjfirub6510MaPBHLmdfMGAcXPIYPHkJLPgOXP4suDx2p1IRoCN0dfiW3AMf/pPqomSqvuh+SVyvw8mvvjWYQPIevvXfLE4qaohQyD5mzEyYczdsKYSF1+nMlz5CR+jq8KyZD2/+FiZcwO5nltPdyUMWws1zptLcbzUnrxjCBWt3RC5nH9L+mqT9J6QxkGfYs+DV4NK7iC7eFcd0hK56b91L8NKPYeSp8LUHu110ywB/OX06O8atJWd3Jje8rcU8EvasT6VqUwr9j2xkwKR6dM2X+KYjdNU7nz0HL8yFvOOCH8K5ErpsaoDbT8/nk+M+IbUhmT/Or+xmHK9CSyhbnY6IIXtCQ/CHYQyI/gTikY7Q1Ve35hl44QfBM0C//TwkpHXZ1AJumT2R1dM/Ja0ujX8+UkNGc+SiKgBh96oMqjcnkz2xAd76XXBJYxV3dISueiR4XNbQb1wjOcfW0VjmYeeCzZjbj+vyMY1u4abL8ygfspH+uwfxj3/vIsmvf/LbQ9i9IgNjQT/ugoZyuOAecHZ/5SgVW3SErnpGDDlTask5to66HYnsfL8/JtD122fVSBdzr02nbPBuRm84grsf1WJuP6FsVQac/v9g7Xx46hKdpx5ndISuDq2piqGnVJE6xMueDSmUrwnOluiMJbDghEReONWP5UtmZuFwrlm6NrJ5VTeEDXMfIWNkJoOtd/H9ZiTFH2bhqz9wpK4zYWKTFnTVvV2rYcEVJOd4KV2RQc3mrueZ16TAny5OYfsQL1btBH78qsXpO7SYR6Parcm0NjrJPbGaEWdVUro0k/pd3V+AREU/Leiqc1YAlt0fvLJ8ag7b38mmparzsw0tgedPdPDCiS78DoukbWdwy2ufMrquLLKZ1VfSVJ7A1kUDyDu5irxTqqku8lL2aTrGr0diY5UWdAUceDKKO9XPkOk1JA/wUb8rgdJlPgK+zot5cX+475xEvhzqp7VuPBPXDec3H75Jir8lUtHVYfA3O9n+TjYDjq6j35GNpOR4gwt8qZikBV3tI05D/yMb6D++AWNBydJMarcl0dnx8oDAa/ku5p/mpFU8UHwmN7xXTEHxQp1jHmOMJZSvyaC+JJEhx9cw4sxKePlaOOP3kDrA7njqK9CCrsCySBvazMBj6vCkBKjbmUjZ6gz8zc6DmjYkwmvHOVh0TAINqa34G0Yz6ZPx3LDiTfp5620Ir0KluSKBrf8dQPbEevo75sP6V+C0X8FxV4Fbj6/HAi3ofZkVgPUvwQd/I++kalqqXWxf1p+m8oPP+tzVD96Z7GDRVBetbgt/43D6bRzH9e+vZfKe5yOfXYWF5XdQviaDmi2t5EypI9X7W/wv/w+V61Op2ZyCsYJ/f+ksmOikBb2PaH+M3OGxyBzRROaYRhLSA3hrXVSuz6RuR9IB67E0JMKS8cK7k1xsGWLACK31E+m3YTLfWrOW04oX4tS1QeKSr97Nzvf7kzTAy4Cj6xk0tY7siQ3UbEmmuijZ7niqC1rQ+wqHISXHS/qwZtKHNuNwQVOlm+KP0qkvTtxXyL0uWDNKWHaEgxVHOGlJsLC8/fCVTWfk9v5cvGEVJ5U8jkMLeZ/QXJHAjncTSB7gJeuIxuBnLEc2wH8uhqMvhiPP63bpBxVZWtDjWWMlbH0fit7hiAt34/QYAj6hdlsy1UUpeGuCJ5P4HfDFMOHtSR5WjDO0ui2MPxF/4zicO46jYNNuztu2jJF1pTZ3SNmlqSKBpooEXMl+skY3kZ31Bbz4Q3AlwugzYEzbLWuE3VH7NC3o8cIKQOUmKPkUSj4JXiS47LPgvsQM6nclUr8zicbdCfhEWJ/n5OP8VNYNh4p+LViuAMZy0Fo3GU/lkRy/uZ5Ti9dxTMWDuE3A1q6p6OFvclHxWTrZz6yF4uXBVTc3LYKNrwUb9BsFQ6dD7lTIy4eco3S9mAjqUUEXkVnAnYATeMgY85cO+6Vt/7lAE/A9Y8zqEGdVfh80lkNdKVRtabtthj2bsXZ8gsMVPAxitQqNVW6212ax1spkbWoG2wa7qBhvqE+2aE6tA2crxjRheQfhqJlIXmky04uayS/dysjaZXpsXHVrw4SJ7bYMnrQBpAz2kpKzi6SSZ3AlPh3c5XAHi/yAIyD7COg/BjLyID0X0ofo7JkQO2RBFxEncC9wJlAMrBCRhcaY9e2anQOMbbtNB+5r+xrd9l6Wyxj2LvxvjAXGYMze7eA+s+8+g2UZjLGwCATvtwwGa18byxgwBstqxQR8GL8XE/CB1Yrlb2m7zweBVkzAG9zvrcfy1hFoacDnrcXnbaDVV0err46AtxoaKylc5sUr4HMIdQ4HDeKgxiRQZRKpyhpDhSRS7nbSkODHGtOIOAJAAKjCWC6Mrz8ebzKZpdmM3JVE/pZmJpbvZnDTEp07rg6D4Kt346t3U70pFTC4UwIk9mslMauVhPTteNI340l9Bel4EmpSP46TVNiSB0lZkJS5/6snNVjw3cnBQzvu5LbtxOC2w9V2c7b7vpPtfS8q+9eBj9P14HsyQp8GFBljtgCIyHzgAqB9Qb8AeMIEK95SEckUkcHGmJAfdF296N8ct+TnNL0H0laEBYOwf5u27f379rYDh+wfeX6YlMjPBma3PWLvTTBCu2dqd7PjTeBpu6Wld7rbGAdYbsRy4fCDx5tA1p400huzyaqHnFrDiPJWxpfVMLCxREfeKgKE1kYXrY0u6ne2G4E7DO7kAO7kAK62r+6kFrwTB8GKlTg9Fs4EC6fb4HBH8n3artB3+n37fwQ6+b4XRgw5DwoKev34rvSkoOcCO9ttF3Pw6LuzNrnAAQVdROYCc9s2G0Rk41dKu182UNnLx7ZTB5Qf/tNERoj6HEOuu0773Cfs7oN9fiib7z/U2z4P72pHTwp6Z/8MdfznsydtMMbMA+b14DW7DySy0hiTf7jPE0u0z32D9rlvCFefe7KsWjEwtN12HlDSizZKKaXCqCcFfQUwVkRGiogHuBRY2KHNQuC7EnQ8UBuO4+dKKaW6dshDLsYYv4hcBywiOG3xEWPMOhG5pm3//cDrBKcsFhGctnhl+CIDIThsE4O0z32D9rlvCEufZe/0PKWUUrFNL02ilFJxQgu6UkrFiZgv6CLyCxExIpJtd5ZwE5G/isgXIrJWRF4UkUy7M4WLiMwSkY0iUiQiv7Y7T7iJyFAReU9ENojIOhH5id2ZIkFEnCLyiYi8aneWSGg76fK5tt/jDSJyQiifP6YLuogMJbgkwQ67s0TIW8BRxphJwCbgNzbnCYt2y02cA0wALhORCfamCjs/8HNjzHjgeODaPtBngJ8AfelqGXcCbxhjjgQmE+K+x3RBB/4J/IpOTmKKR8aYN40x/rbNpQTn+8ejfctNGGN8wN7lJuKWMaZ074J2xph6gr/oufamCi8RyQPOAx6yO0skiEg6cCrwMIAxxmeMqQnla8RsQReROcAuY8wau7PY5PvAf+0OESZdLSXRJ4jICOBYYJnNUcLtDoIDMsvmHJEyCqgAHm07zPSQiKSE8gWiej10EXkbGNTJrt8CNwNnRTZR+HXXZ2PMy21tfkvwT/QnI5ktgnq0lEQ8EpFU4HngRmNMnd15wkVEZgPlxphVIlJgc5xIcQFTgOuNMctE5E7g18D/hPIFopYxZmZn94vI0cBIYE1wKXbygNUiMs0YszuCEUOuqz7vJSJXALOBM0z8nkTQJ5eSEBE3wWL+pDHmBbvzhNlJwBwRORdIBNJF5D/GmG/bnCucioFiY8zev7yeI1jQQyYuTiwSkW1AvjEmrldsa7vQyD+A04wxFXbnCRcRcRH80PcMYBfB5ScuN8asszVYGLVdJOZxoMoYc6PNcSKqbYT+C2PMbJujhJ2ILAauNsZsFJHfAynGmF+G6vmjeoSuDnIPkAC81faXyVJjzDX2Rgq9rpabsDlWuJ0EfAf4TEQ+bbvvZmPM6/ZFUmFwPfBk27pYWwjxMilxMUJXSikVw7NclFJKHUgLulJKxQkt6EopFSe0oCulVJzQgq6UUnFCC7pSSsUJLehKKRUn/j8y1ill371sXQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_normal(n=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d51f7e14",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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nAXuNMQXGmFZgEXDFcW2+DSwxxhwEMMaU+bZMpaxX11oHwModTi4Zn0FkmL4ZqvxLVwI9Cyhqt13cdlt7I4BEEckXkfUi8l1fFaiUvzjUcAjBRqMjhGtysq0uR6kTdGVxLungNtPB40wBzgcigVUistoYs/uYBxK5GbgZID09nfz8/FMuGKChoaHb9w1U2mdrtXpaWXJoCeGOMaRFhdKwfzP5hR39apwef+pzb9E++05XAr0YaP/uTzZQ0kGbCmNMI9AoIh8DE4FjAt0YsxBYCJCTk2Py8vK6VXR+fj7dvW+g0j5b6/kvnqexqJGm0uncfvZwZs4c3iPP40997i3aZ9/pypDLWmC4iAwWkTBgHrD0uDZvADNEJEREooBcYIdvS1XKGrUttTyx6QkGRZ6Ju2kI35h0/IijUv7hpEfoxhiXiCwA3gHswF+MMdtE5Na2/U8bY3aIyApgC+ABnjPGbO3JwpXqLcX1xbg8LiqOjGH6kBS9KpHyW126wIUxZjmw/Ljbnj5u+3fA73xXmlL+YV3pOgCOlKVz1zf0zVDlv/RMUaVOYn/tfuyEE2lLYs64flaXo1SnNNCV+hpVjire3Pcm7vpxzBmXQXS4XrVR+S8NdKU6YYxh8a7FtHpaaaqcyjVTdLhF+Tc93FCqE89seYYnNz1JnGciseEjyB2cZHVJSn0tPUJXqhOrSlYxMmEMJbuv5ZopA7HZfH8ikVK+pIGuVCeONB7B05qCMTaunqxzz5X/00BXqgNlTWWUNJZQUp7ItEFJDEyOtrokpU5KA12pDqwqWQVAWelAfTNUBQwNdKU6UFhXiGAjzJPJnPE691wFBg10pTpQ7ajFuCOZMy6L2IhQq8tRqks00JXqwJ7ycjzuCK6erMMtKnBooCvVgcLqCkKIZvrQZKtLUarLNNCVOk5pnYOa1jIyY9Ow69xzFUA00JU6znNrPsUWXsbMgTlWl6LUKdFAV6qdZQXLeOXQvdhMFDdO/JbV5Sh1SjTQlWrn16sfxeVI5YfD/pfUqFSry1HqlGigK9XmpR0vUeesxlOXy3VTJlhdjlKnTANdKaCwtpDH1j0GTWO5IPty4iN17rkKPLp8rlLA2/vfxu1x01B8JdfM6g/AjlGjT2g3eqde+1z5Lz1CVwrYVrkNu4khLTqFGcN17FwFJg101edtKd/CR8Uf0VQxnSsnZenccxWwNNBVn/fnjX8m0hZPS9XZXKOn+qsApoGu+rS91XtZfXg1oQ2zmJiZxvD0WKtLUqrbNNBVn/b+wfcBKCkZruueq4Cnga76LGMM/9z5T/qFTiLUk8JlEzOtLkmp06KBrvqsI41HqHJUUVY6lAvGpJEQFWZ1SUqdFg101WfVtdYB0Nik656r4KCBrvqs+tZ6AOLCojl3hM49V4FPA131WVtKdwFw8YgJhNr1V0EFPv0pVn2SMYY3dn+AxxXF/KkTrS5HKZ/QtVxUn/Ll+iyfjRYKrrQTW5KLzDoTTnONFl33RfkDDXTV57gFXjwvDLcjlctXW12NUr6jQy6qz/l0jFCR6MZVNouZRZusLkcpn+lSoIvIbBHZJSJ7ReT+r2k3VUTcInKN70pUyncMsORsGzSlMXkPJLQ2Wl2SUj5z0kAXETvwJDAHGANcJyJjOmn3KPCOr4tUyleKU6AkWXDUnsOFReutLkcpn+rKGPo0YK8xpgBARBYBVwDbj2v3I+B1YKpPK1TKhzYP9i6NG1GVxdQjrx/dUV0Ih9ZDxR5obQB7GInDGmmuCsVRHQpGl9RV/k+MMV/fwDt8MtsYc1Pb9nwg1xizoF2bLOCfwCzgeeAtY8xrHTzWzcDNAOnp6VMWLVrUraIbGhqIiYnp1n0DVbD32bFt2wm3uQYO7HKfO7p/R/4a9h5bpZwzj9zGvIRSspvXkuXYQJz78Fdt3LYwbB4XggeAVonkcMQZHIw4E/fEvC4/f8TYsV2qqb1g/z53RPt8ambOnLneGJPT0b6uHKF3dGhy/KvAn4D7jDFukc6PZIwxC4GFADk5OSYvL68LT3+i/Px8unvfQBXsfd5x6w9PuK306ae63OeO7n+81SOF7VfZcVfncufGXzNy+CFCIjw0VYTCd34Ng2ZA6kjsIeHg8bAnZxSRya3EZjfTP3sVAxpXIbXlcP7PIebYM0s7ev7uTFsM9u9zR7TPvtOVQC8G+rfbzgZKjmuTAyxqC/MUYK6IuIwx//ZFkUqdrqZweHa2jYiGZF4sX8PY8UU0loZR/EkczZVhjH7itmPvYLPharJT3xRJfVEk9nA3yaMbSA5dDDuXwyW/h3FXWdMZpTrRlVkua4HhIjJYRMKAecDS9g2MMYONMYOMMYOA14D/q2Gu/MnObKE+SvhNXQEj5TCHVidw8MNkmiu7tsKiu8VO2aZ4uOUTSBwEr90Ib98PblfPFq7UKThpoBtjXMACvLNXdgCLjTHbRORWEbm1pwtUyhfKRjsBiHHEs+e9dOoKo+h4NPEk0kbBD96DM2+DNU/BS1dDS71vi1Wqm7p0pqgxZjmw/Ljbnu6k7Q2nX5ZSvmKoOLuZl8fGMMBh54Xdc7mt9l+n95D2EJj9a0gfA0t/DC98A1uoB49Tz9NT1tKfQBXUksbV83huBFFOYefBe5hVuMF3Dz7pevjW36FkEwNmVmIL9fjusZXqBg10FbQShzfQcoaDXeFhuIvPIavawajqA759ktGXwbx/EpHgJHtGFWL7+mnASvUkDXQVlOIGNNFvSh3FlREAlISO4OIDn3dn1PzkRlxEyZoEotNayZxeDaKhrqyhga6CTkRiKxm5NTSVhXFoZxwAdreNWT14qn/dgSiObIgjrr+D1PH6Jqmyhi6fq4JKSISb7BlVuBx2DnyWyJuX2cHAGSVlPb4QV/XuaMLjXKSMafAuF6BUL9MjdBU8bIasc6qwhxmKP0lia3oIG4cJjtJLuXxn15YGOD3CkQ3xNJWHkZlbA4e39MJzKnWUHqGroJE2oY6oFCfFnybSUhNK4XDv7UlHBjCp7I2vvW9HVxzqVjuPULwykcEXl2N79Qa45SMIj+3SYyt1uvQIXQWF6AwHyaMaqd4TRX1RJAA7M6LwuGK4ZM8WbCcsP9Rz3C12SlYlQvV+WH5Prz2vUhroKvDVl5KZW4OjJoTSTfFf3bw3NQ6csVx4cG2vl9RUHg7n3gubX4bN3VtVVKlTpYGuApvHA/+6BVuI4dBniRi3d2LijsxQKlOqSauMsO6qROfeAwPOgrfuhKoCa2pQfYoGugps656Hgg8p3RhHa93RmSV/vjQS447mux86rKvNHgJXPwu2EPj3bd4XH6V6kAa6ClxV++G9n8KwC6jZF/XVzS6BisQWosqHkltcZGGBQHw2zPkNHPwM1nS4/JFSPqOBrgKTxwNvLPAe/V72OO1XTvz35FSwuZlyoLpnzgw9VROvgxGz4f1fQMVeq6tRQUwDXQWmtc/BgZVw8a+8R8FtHKHw2qw6TGMW31vt43VbuksELv0ThETAv38IHrfVFakgpYGuAk9VAfznZzDsApg0/5hdKyaF4wlxM257NgmOVosK7EBcBsz9HRR/DqufsroaFaQ00FVgMQbe+snRoZZ217D9fITwz1luXA0juH6tHw5tjP8mDL8YPvwV1By0uhoVhDTQVWDZ+joU5MP5Pz1mqKUlBJ681IY0ZTBiYy5Da0utq7EzInDJY96vl93tfXFSyoc00FXgaK6BFQ9A5mTI+f4xu+ojoTlcaK6bzlV7VllTX1ckDICZD8Ged2D7v62uRgUZDXQVOD54BJoq4NI/gs1+zC5X22ZCcyM5pbssKO4U5N4KGRPh7fu8L1JK+YgGugoMxeth7fMw7RbIPOOE3Rsz0wE4+2Bhr67b0i32tvH/xnLvVEalfEQDXfk/twveuh1i+8HMBztskj9wFADnFe7uzcq6L3OS90h93V/g4Bqrq1FBQgNd+b/PF8KRL2D2byAi7oTdJTXN7Ej2vkEa0+rs7eq6b+ZDEJcNb94O7gCqW/ktDXTl32oPeaf5DbsQxlzRYZO/f1aI2L1rtoS5erO40xQe452bXr4DVj1hdTUqCGigK/+24n7wuLzBJyeeyF/b7OSlDZ8Tk/ge6dWGpEC7nOeouTDqUsh/FKoLra5GBTi9YpHyW0mV62DHUpj1MCQNBk68YtAf8sZD7h5CW9zc/oabED9a0LCzqxuN3rnj2BvmPIr7t+Novm8ajgk/ZcetP+y4nVInoUfoyi+J3cPwPc9Aykg468cdtmkIDWH15BIiGuP400Inww73cpG+Ep9NxRexxGS2kN661epqVADTQFd+KWVsA5GOMu+c85CwDts8enkahNdy1coQEpp6uUAfq9oTjaM6hNENy7D5058ZKqBooCu/ExbnJHlUA4f7zYJBZ3fYpiDNxq7hZUSXjOfKTSW9XGEPMMLhtQmEe+pJnRBobwQof6GBrvyMISOnFrdTKBhyQ6etXp2ajsHGdz9s9o81z33AURXGwYhpJA5vJCLRj1aKVAFDA135lfjBzUSltVK2OQ5nWHyHbTwIWwZEYm9K5ryDfn6a/ynaE30xboeNflNrvSdUKXUKNNCV37CHuUk7o46m8jBqC6I6bfdJ1jhaY8sYdcjt/6f5nyKXLYIjG+KJTHLC2metLkcFGA105TfSzqjHHurhyLp46GQgxYPw3Gw7Yndw8RflvVtgL6kviqDhcDh88EvviVVKdVGXAl1EZovILhHZKyL3d7D/OyKype3jMxGZ6PtSVTCLTG0hYUgTlbtiaKkN7bTdm6NG0pK2lYnbk5m+O7iOzo8S74uaxwUr7rO6GBVAThroImIHngTmAGOA60RkzHHN9gPnGWMmAI8AC31dqApirlYycmppbbRTsTWm02aFtQd5+eIqMMINH5f1YoG9z9kYAufeAzvehF0rrC5HBYiuHKFPA/YaYwqMMa3AIuCYRTWMMZ8ZY6rbNlcD2SjVVaueIDzeRen6eIy74x/JfTX7uH7ZDbjDHFz57gCyqoP16Lyds34MqaNg+T3Q2mh1NSoAiDnJZbBE5BpgtjHmprbt+UCuMWZBJ+3vBkZ92f64fTcDNwOkp6dPWbRoUbeKbmhoICam8yO5YOQvfXZs29bh7RFjx3br8SKaS5m6dgHlIcPYFH/9MftcAwey37af/Pp8djt2gzuS6EM38NNEgy2A5yp29H/15f9ra1oaYWVlX7WLr9nGpE0PcrD/VRQM/V6v1tlb/OVnuzedTp9nzpy53hiT09G+rqzl0tGvToevAiIyE/gBcE5H+40xC2kbjsnJyTF5eXldePoT5efn0937Bip/6fOX64wcr1vrjhgD//wW2EOpf62SgU3Hrji46LffYknlEjKiM5iRPJ+3Vw3grpWLGVwa2GucdPR/9eX/64EfLWDgn59o1y4PZDsDNi9iwKV3Q3r3Xjj9mb/8bPemnupzV4ZcioH+7bazgRNOzRORCcBzwBXGmErflKeC2hevwZ53YdZDuJqOvaTcx+OEJdVLuGDABbwy9w3WbJxE7sCBTAvwMO+WCx+BiHh48w7w6LIAqnNdCfS1wHARGSwiYcA8YGn7BiIyAFgCzDfGBMglY5SlGsrh7XshK8d75Z52PMCzF9sYFj6MR899lOc/OUhlYysPzh0dNGeFnpKoJLjol1D8OWz4u9XVKD920kA3xriABcA7wA5gsTFmm4jcKiJf/ib+FEgG/ldENonIuh6rWAWHt++Flnq44okTLvjcHA4tYcL4qPFUNXh4bmUBl0/MZEJ2gjW1+oOJ18GgGfCfn0FDcM/wUd3XpXnoxpjlxpgRxpihxphftd32tDHm6bavbzLGJBpjzmj76HDAXikAdi6DbUvgvHsh7cQ1wxsjvJ8jbZE89u4uPB645+KRvVyknxGBS/4ArU3wzkNWV6P8lF7gQvWu5hp4605IHwfn/KTDJoVp3oGV+qYoXttQzC3nDaF/UudLAQSazi58cVKpI2DGnfDRoxQ9+R8aSiK+2qUXw1Cgp/6r3vbuQ9BY7h1qsZ94RmhTOPz1IhtZ5YaVewfTLy6CH88abkGhfmrGXTiqQ8iYWoM9zG11NcrPaKCr3rNzOWx8Ec7+MWRO6rDJq+fYqIqBiRvHcrDOzn9dOprocP1D8ish4ZSsTsQe5qFfTi2dzCBWfZQGuuodDWWw9EfQbzzkPdhhk8+HC+9OEs7cbmdZ+jWMTrJxyfiMXi7U/7XUhlK+NZa4AQ7iBjZbXY7yIxroqucZ4w3zlnq46tkOLyn3xUDhsWvsDCiH5opLaA4JZ/6YcET65ETFk6rcGUNTRSj9ptQSEqlDL8pLA131vPV/g90r4MJfdDirBWDlWCG62XDJe8P4JP1s5u1+n8wY/fHslBFKViciNsjIrdYTjhSgs1yUj3Q0c2P0zh1QsQfeeRCG5MG0Wzq9f0Uc9KsWnh7/LQbVHubaXe9TxTd6sGL/d7LZMM6GEEo3xJExrZayawZQuT32mP0686Xv0UMg1XNam2DxdyE0Eq58Cmyd/7jVRAuVIelUR8Tyk42vEGp0GKEragqiqD0QSeq4eiJTW6wuR1lMA131nOX3QNkO77h5XGanzdYdWUdRmnDEM5Wr9+Qzoqa4F4sMdMKRtfE4G+1kTa/WqYx9nAa66hHxg5tg04veizQMO/9r27626w1wh5F9oD/X73y3lyoMHh6XjUOfJWIP95CRW4NOZey7NNCVz4XHO+k3pRYGnwt5J1yx8Bi7q/bw3v5P8TgyuP/zVwjz6JXuu8NRHUbZxnhis1pIGVdvdTnKIvqmqPIpe7ib7BlVuFsF29XPn7DwVnu/X/d7/rbtbxhPGLM2RTNAF506LdV7o4hIaiV1XAMtNZ1fl1UFLw101alTXnPEZsg+p5qQCDcHPkjBkXPeCU2+nHlR5ajihe0v4qofw4yE2/jhxyfOgHFs29bpBTVUR4Qj6xIIi3ORmVsDpdtO64IYnX3/dfaM/9IhF+UjhoycWqJSWylZk4ij6sSThwCcHifvH3yf7y7/Pm439PNcxh+/eVbfXOe8BxiPcGhlEm6nwMvX6VK7fYweoSufSB7TQMKQJsq3xlBfFHnCfqcdluYK/3n1Qiodldg88VB2I3/9wTd0rRYfcznsFK9MYvAl5fDSN+GGZRDet67Z2VfpEbo6bQlDG0mbUE9tYSQVW2NP2H8gFe670c4r59kZkzyG4ebH1O2+lz9fOY9BKdEWVBz8HFVh8M2/wZEvYPF8cLVaXZLqBRro6rTEZjfTb0otDSXhlKxJ4Phrim8eLDw8305jBNz/ipu42lvZsDOT/75iInkj0yypuc8YcTFc9jjs+wCWLtDlAfoA/VtXdVt0hoPM6dU0V4ZS/GkimKNhboCNQ4U/XmmjXzU8sNjNsuyLWLS2iAUzhzH/zIHWFd6XTJ4PDaXwwSNgC4XL//y1Z+yqwKaBrrolJtNB9tlVOGpDKfo4GeM+GhI7s+FvF9gpyBDSqw0PLHbzTub5/HPURVx44HPm3n43Ok+iF517N7id8NFvANhx//sc/5eUzlwJDhro6pTFZjeTdVY1jqpQDn6UjMd5NMwPpsKvv2UnthluWe7m3K2G14aezz/GzGFW0Xpu3/iqzmixwswHvJ8/+g0ZuZEc/jzhmL+oVHDQQFenJGFYI/0m19JcGUrRR8l4XEfDvDgZHr3GTmQrPPKCm8QG+Pvo2bwy8gLOP7iOn2x4Bbuelm6dmQ+AzU4CvyIk3EPxZ4kYlw6/BBP9bqouMqRNrCUjp5aGw+EczD8a5lsHCI/Ms3HnzSHUR8J9r7mJbbTz+8nzeGXkBcwuXK1h7i/Ou5fDa+OJ7tfCwFmV2MN1Ma9gokfo6qRsoR4yptUQ199B1Z4oSjfEH/Pn+jNzbTjC4Lp8N+dvMuCK4uGz5rM5dTjf3f4283a/r8MsfqRmXzSuZjtZZ1Uz+OJyilcmWV2S8hE9QldfKzzeyeCLyonNclC6IY7S9ceG+cFUKI+HWZsM31hlOBzenx/N/AnbkgZz1/qXuU7D3C81lERQ+J8UjEcYeH6F96pSKuBpoKuOeTwkDGtk0IUVSIjhwAfJVO2O4fjZEctzbIS3wpy1hjeGnMPdM24DDL//5AkuKFpvSemqa1pqQtn/bipNpeHw5u3w6g3QVGV1Weo06JCLOlH1AVi64Kvx8pLVCbhbTlw10S2wfYCQXR7C/0y+kS2pw5h2ZDt3rV9EnLPJgsLVqfK02ij6JInRC/8PfPg/cOAz71z1ERdbXZrqBg10dZSrBVY/BR//DoDDa+Op2RfF8UflAB6Bp+faOZIkFNdfgyRkccfGxVx04HMdYgk0RmDGXTDsQvjXLfDPb8HoywiJcuFq0ogIJDrkosAY2PEWPJkL//kZDJoBP/yMmn3RdBTmBnj+nGw+miC0lM9iXEEYT3/wGBdrmAe2jAlwcz7Mehj2vs/QuWWkjKvDFqJLBgQKffntyzwe2PmW94j8yBZIHQXXL+n0knEehFUZY3lpwjTKxi0lpDmRh5fuJ6dMLxsXNELCvWeWTpxH/R1TSB3XQNLwRip3xVC9O/qY8w6U/9FA74sctbD5FVj7HFTsgqQhcMWTMOFasJ94pZuG0Ag+yJ7CssHTKUoOIbb/M0SYJn76ipMRZeUWdED1uPhsSlYlUbWzlZSxDaRNqCd5dAO1+6OgfDekjrC6QtUBDfS+wu2E/R/DtiWwdQk4myBzElz1LIy9ih1jxwO/+qq5U+xsShvFx9kT+SRzIi0hoWSHvE9y/w8Jc7t4+GU3g/TaCUGjs6sTOarDKF6ZRERiK0kjG0kY2ghPTqWxNIy6A5HUF0fibrWd0lowxz+X40cLIC/vdMpXbTTQg1ljJRR+Anvfg53LoLkawmJg3FWQ8wPImnxM8+rwGDanDGNd+ihWZ4ylMTSSSKeDWUXrSQ//kNfPr2NkseGHy9xk6uy2PsVRHUbJ6jDsG+NIGNpE/KAmMqbV0i+nlsbScO+b6UNnQcoIEH0nxSoa6MHC44aKPXB4E5RsggMr4chWwEB4HIycA2OugKHnQ2gErS4Puw/Vsr2kji8O1fLRrLs5GNcPgJjWJqYf3so5h7Ywom4PK8e7eWuajdQa+MWLbmx6Bn+f5W6xU7k9lsrtMYQnuIgb0ExsdjOsuN/bIC4LBp4FWVMgc7L3jdbQE69gpXpGlwJdRGYDjwN24DljzG+O2y9t++cCTcANxpgNPq5VuZ3QWA51h6FqH1QVQOU+79dlO8HZ6G0XEglZU2g9934q06ZzIHwkhdWtHChs4sD6beyvaGJvWT1OtzeZY8JDGNVcwwUH1zGhYh/Dag9hNx6awuH2m+3UxtgZfdDw7XwNc/UloaUmlPKaUMq3xDF61Qoo+BD2feidy/7Fq23N7JA0GJKHQ8owSB5OVHoLriYbrma7vsnqYycNdBGxA08CFwLFwFoRWWqM2d6u2RxgeNtHLvBU22f/Zky7z6btSw8YgzFfbnv3ma9uM23NDQbvdC7jOXp/TNt9jMEYl/fSX24nxt2KeJwYV4s3mN2t4HF65347HdBSj8dRi8dRD446TEs9tNRjd1Rjbyont/4w5NcdWz5CfVg6FeHZHIqdzV77ULZ4BrO5OY3S/S4ad7mBBsB7xmaoXeifGMWA5CjOG5HK2Mw4xmXFMyAxkrVT7+BQMhQNElYnCyXJNopShNoY4d5X3eTs1SRXXyNxIEy5wfsB3oOOkg1QshHKd0HlXu+Vk9wtDJx59G5up9C/6o/wl7cgIh4iEyAi4ejXoVEQEgGhEd4Dlfaf7eFgs4MtxPtZ2n391XbbbWLvExf26MoR+jRgrzGmAEBEFgFXAO0D/QrgH8abeKtFJEFEMowxh31d8IZ3XmDqZ3fR9CFIW4gKBuHoNm3bR/d92Q5s8vXBJMd9tkKTCaeBSOpNJHVEU2YSKDfZlJsEykigzCRwwKRTZNJwtoQR0xpCbEQo0eF2kqPDGZcSTl5MOKmx4aTEhJERH8nA5Cgy4iMIsR/7Q13tqObcxRdRd/vRH4UwpyGzEkYcMsxd62GKhrk6VXEZEHcJjLrk6G0eN9QWceCKWYREugmJdBMa6aF1fBoxNjvUFkPpNnDUQEtdpw/tO+1+y+W43/zT3T6JQZmX9MgbwfLlkWinDUSuAWYbY25q254P5BpjFrRr8xbwG2PMyrbt94H7jDHrjnusm4Gb2zZHAru6WXcKUNHN+wYq7XPfoH3uG06nzwONMakd7ejKEXpHLznHvwp0pQ3GmIXAwi4859cXJLLOGJNzuo8TSLTPfYP2uW/oqT53ZVCpGOjfbjsbKOlGG6WUUj2oK4G+FhguIoNFJAyYByw9rs1S4LvidSZQ2xPj50oppTp30iEXY4xLRBYA7+CdtvgXY8w2Ebm1bf/TwHK8Uxb34p22eGPPlQz4YNgmAGmf+wbtc9/QI30+6ZuiSimlAkPwT8xUSqk+QgNdKaWCRMAHuojcLSJGRFKsrqWnicjvRGSniGwRkX+JSILVNfUUEZktIrtEZK+I3G91PT1NRPqLyIciskNEtonI7VbX1BtExC4iG9vOZQl6bSddvtb2e7xDRKb78vEDOtBFpD/eJQkOWl1LL3kPGGeMmQDsBh6wuJ4e0W65iTnAGOA6ERljbVU9zgXcZYwZDZwJ3NYH+gxwO9D1tXcD3+PACmPMKGAiPu57QAc68EfgXjo4iSkYGWPeNca42jZX453vH4y+Wm7CGNMKfLncRNAyxhz+ckE7Y0w93l/0LGur6lkikg1cAjxndS29QUTigHOB5wGMMa3GmBpfPkfABrqIXA4cMsZstroWi3wfeNvqInpIFlDUbruYIA+39kRkEDAJWGNxKT3tT3gPyPrKRUuHAOXAX9uGmZ4TkWhfPoFfr4cuIv8B+nWw6yHgQeCi3q2o531dn40xb7S1eQjvn+gv9WZtvahLS0kEIxGJAV4H7jDG9MYKVZYQkUuBMmPMehHJs7ic3hICTAZ+ZIxZIyKPA/cDD/vyCfyWMeaCjm4XkfHAYGCzdyl2soENIjLNGHOkF0v0uc76/CUR+R5wKXC+Cd6TCPrkUhIiEoo3zF8yxiyxup4edjZwuYjMBSKAOBF50RhzvcV19aRioNgY8+VfXq/hDXSfCYoTi0SkEMgxxgT1im1tFxr5A3CeMSZor84sIiF43/Q9HziEd/mJbxtjtllaWA9qu0jM34EqY8wdFpfTq9qO0O82xlxqcSk9TkQ+AW4yxuwSkZ8D0caYe3z1+H59hK5O8AQQDrzX9pfJamPMrdaW5HudLTdhcVk97WxgPvCFiGxqu+1BY8xy60pSPeBHwEtt62IV4ONlUoLiCF0ppVQAz3JRSil1LA10pZQKEhroSikVJDTQlVIqSGigK6VUkNBAV0qpIKGBrpRSQeL/A9FRkAQFC+sbAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_normal(n=200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5a3d8bb8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_normal(n=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f3b065f",
   "metadata": {},
   "source": [
    "## Sorting data that comes from a distribution\n",
    "\n",
    "The best sorting time for arbitrary data: $O(n \\log n)$ (quicksort)\n",
    "\n",
    "For data that is sampled from a fixed distribution (Independent Identically Distributed or IID) we can sort in $O(n)$ time!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5600d268",
   "metadata": {},
   "source": [
    "### Easiest case: Uniform distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6187ce82",
   "metadata": {},
   "outputs": [],
   "source": [
    "n=10000\n",
    "R=array([random.uniform(0,1) for i in range(n)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "bae6e9d7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(R,density=True);\n",
    "grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "039bd984",
   "metadata": {},
   "source": [
    "### Create a list of lists\n",
    "* Size of (outside) list is $n$.\n",
    "* Each inside list starts empty.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "cc557420",
   "metadata": {},
   "outputs": [],
   "source": [
    "m=n #int(n/10)\n",
    "L=[[] for i in range(m)]\n",
    "\n",
    "for r in R:\n",
    "    i=int(r*m) # scale random number from [0,1] to [0,n]\n",
    "    L[i].append(r) #append number to list at location i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "dbac8739",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[], [0.0001494277885513151]]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L[:2] "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "246f2207",
   "metadata": {},
   "source": [
    "### Size of fullest bin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "bb5bf8d2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 8)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lengths=[len(l) for l in L]\n",
    "min(lengths),max(lengths)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "894b8e3e",
   "metadata": {},
   "source": [
    "### sort each short list and concatanate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "188e7689",
   "metadata": {},
   "outputs": [],
   "source": [
    "_sorted=[]\n",
    "for l in L:\n",
    "    _sorted +=sorted(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "6d3ffeb8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0001494277885513151,\n",
       " 0.0007499316913494036,\n",
       " 0.0014788538473297086,\n",
       " 0.0016874161325813875,\n",
       " 0.0018594589400154904,\n",
       " 0.0019749541714617624,\n",
       " 0.0020417289951969453,\n",
       " 0.002168231059543224,\n",
       " 0.0021831743262641368,\n",
       " 0.0022116091240896862]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_sorted[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "fede5e3b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resort=sorted(_sorted)   # checking that the order is good.\n",
    "resort==_sorted"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "747afa2b",
   "metadata": {},
   "source": [
    "### What about distributions other than uniform\n",
    "We can use the CDF of a distribution to transform it into a uniform distribution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "48235770",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "eb2e0218",
   "metadata": {},
   "source": [
    "### Transforming the distribution to a uniform distribution\n",
    "We use $F$ to denote the CDF\n",
    "* $F(x) = P(X \\leq x)$\n",
    "*\n",
    "* $X$ is a random variable, therefor $F(X)$ is a random variable.\n",
    "*\n",
    "* What is the distribution of the RV $F(X)$ ?\n",
    "* $0 \\leq F(X) \\leq 1$, Therefor $P(0 \\leq F(X) \\leq 1) = $ ?\n",
    "* What is the probability that $0\\leq A \\leq F(X) \\leq B \\leq 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "96b961a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#figure(figsize=[15,10])\n",
    "mu=1; sigma=1; n=10000; m=100\n",
    "s = np.random.normal(mu, sigma, n)\n",
    "\n",
    "xmin=-4; xmax=6; delta=1/n\n",
    "x=arange(xmin,xmax,delta)\n",
    "cdf=norm.cdf(x,loc=mu,scale=sigma)\n",
    "plot(x,cdf)\n",
    "grid()\n",
    "plt.hist(s, 30, density=True);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed9bee1e",
   "metadata": {},
   "source": [
    "* $F(x)$ is a non-decreasing function from $(-\\infty,+\\infty)$ to $[0,1]$. Therefor for any $0 \\leq A \\leq B \\leq 1$ there exists $a \\leq b$ such that $F(a)=A$,$F(b)=B$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87c54d2b",
   "metadata": {},
   "source": [
    "* $P( a < X \\leq b) = P(A \\leq F(X) \\leq B)$\n",
    "*\n",
    "* On the other hand $P( a \\leq X \\leq b) = F(b)-F(a) = $\n",
    "*\n",
    "* Therefor, for any $0 \\leq A \\leq B \\leq 1$:  \n",
    "$ P(A \\leq F(X) \\leq B) = B-A $\n",
    "* \n",
    "** Which implies that the distribution of $F(X)$ is ? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "7f55176b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "index=np.array(x.shape[0]*(s-xmin)/(xmax-xmin),dtype=np.int)\n",
    "scaled=cdf[index]\n",
    "hist(scaled,density=True);\n",
    "grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f1aeb46",
   "metadata": {},
   "source": [
    "### We can now use the sorting method for the uniform distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "8eda355a",
   "metadata": {},
   "outputs": [],
   "source": [
    "m=100\n",
    "L=[[] for i in range(m)]\n",
    "\n",
    "for j in range(s.shape[0]):\n",
    "    r=s[j]\n",
    "    _scale=scaled[j]\n",
    "    i=int(_scale*m)\n",
    "    L[i].append(r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "3bf325e3",
   "metadata": {},
   "outputs": [],
   "source": [
    "_sorted=[]\n",
    "for l in L:\n",
    "    _sorted +=sorted(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "71550cfb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-2.519609128168893,\n",
       " -2.3544165734214726,\n",
       " -2.3470217906548867,\n",
       " -2.1142300462692667,\n",
       " -2.1085060675418363,\n",
       " -2.0771503919752417,\n",
       " -2.053087616509429,\n",
       " -2.053028113499684,\n",
       " -2.0053252271060855,\n",
       " -1.996303937273491]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_sorted[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "0141acf4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resort=sorted(_sorted)   # checking that the order is good.\n",
    "resort==_sorted"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f788b0a",
   "metadata": {},
   "source": [
    "###  Sorting when the distribution is not known\n",
    "When the distribution is not know, we can use the empirical CDF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "d35723fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(100000,)"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "## Lets generate a complicated distribution\n",
    "n=25000\n",
    "s1 = np.random.normal(0, 1, n)\n",
    "s2 = np.random.normal(5, 2, n)\n",
    "s3 = np.random.normal(-10, 5, 2*n)\n",
    "s=concatenate([s1,s2,s3])\n",
    "s.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "0359fd39",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(s,bins=100);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65d5d064",
   "metadata": {},
   "source": [
    "### Calculating the CDF requires sorting\n",
    "So calculating the CDF requires sorting and $O(n)$ sorting requires knowing the CDF ...\n",
    "\n",
    "Are we stuck in an infinite loop?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "14d4bbaf",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_CDF(s):\n",
    "    x=sorted(s)\n",
    "    p=arange(0,1,1/s.shape[0])\n",
    "    plot(x,p,label=str(s.shape[0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "93446fdc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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svVsikAJdRPZbVVRFWWWA8087wetS5Cgo0EVkv8Vbq4iLacHgNJ1ZMRIp0EVkvyVbq/iPU5N07pYIpUAXEQC+27abrXsd53Y/3utS5Cgp0EUEgM+ztwBwfg/Nn0cqBbqIAJCVvZUTWxudOrT2uhQ5SkEFupldZGbZZpZjZpMP0SbDzBaZ2XIz+yK0ZYpIY9pbXsWctYWccbzmziNZdEMNzCwKeBoYDuQBc81shnNuRa027YBngIuccxvMTO/ZRCLInHWFlFUG6JMU43UpcgyCGaEPAnKcc2udc+VAJjCqTpvrgbeccxsAnHNbQlumiDSmL7K3EhfTgh6JGqFHsgZH6MDJQG6t5TxgcJ023YEYM8sCEoAnnXMv1X0gM5sITARITk4mKyvrKEoOLyUlJb7oR7DUX3+atWgP3du2oHzv7mbR33389voGE+j1Hf/r6nmcAcBQoBXwtZnNcc6tPuBOzk0FpgKkp6e7jIyMIy443GRlZeGHfgRL/fWfDYV72DTrc24d2oM2Fet939/a/Pb6BjPlkgd0rLWcCuTX02aWc263c24bMBvoG5oSRaQxffHtVgDO0/7nES+YQJ8LdDOzNDOLBUYDM+q0eRf4nplFm1lrqqdkVoa2VBFpDF9kb6VjYivSdDGLiNfglItzrtLMJgEfAVHAC8655WZ2W832Kc65lWY2C1gCBIDnnHPLGrNwETl25ZUBvlqzTdcO9Ylg5tBxzs0EZtZZN6XO8hPAE6ErTUQa27z1Rewpr+Lcbppu8QMdKSrSjH2+agsxUcbZXZO8LkVCQIEu0kw55/h4xWbOOjWJNi2DerMuYU6BLtJMrSwoZn3hHi7spWuH+oUCXaSZ+mBpPi0MRvY+0etSJEQU6CLNkHOOGYvzOfvUJDq0ael1ORIiCnSRZmjud9vJLdrLlWee7HUpEkIKdJFmKHPuBhJaRjPyDE23+IkCXaSZKSmr5MOlm7i0bwqtY7V3i58o0EWamenz89hbUcW16R0bbiwRRYEu0oxUVgV47su19O/Ujv6d2ntdjoSYAl2kGZm1fBO5RXuZeO6pXpcijUCBLtJMOOd4NmsNaUnxDNfBRL6kQBdpJmYu3cTy/F38KONUolrozIp+pEAXaQb2lFfy2KxVnHZiAlf1T/W6HGkk2mdJpBl4+P2V5G7fw6s/HKLRuY9phC7ic28tyOPVbzYw8dxTGHJKB6/LkUakQBfxsS+/3cbk6UsZnJbILy7s4XU50sgU6CI+9frcXG6aNpe0pHj+OnYAMVH6c/c7zaGL+ExxaQW/nbmSV7/J5ZyuSfzl+jNp1zrW67KkCSjQRXyioipA5jcb+POn31K0p5zbzjuVu0f00IegzYgCXSTCOef4dOUWfvfhStZu3c3gtETuu6QnfVLbeV2aNDEFukgEm/ddEb/7cBXz12/nlKR4nhuXztCeJ2CmUXlzpEAXiUAFO/fyxKxs3lq4kaQ2LfndVWdwzYBUffDZzCnQRSJIaUUVz3yew9R/rqUq4Jh0flfuOL8rrWKjvC5NwoACXSRCzF9fxL3Tl5KzpYTL+p7EPSN60DGxtddlSRhRoIuEufLKAH/6dDVTvljDSW1b8dJNgzi3+/FelyVhSIEuEsbWF+7m9pcXsKJgF6MHduRXl/aiTUv92Ur99JshEqZmLM5n8vQlxES14G/j0nUOc2mQAl0kzFRWBXjo/RW89PV6BnRuz5Oj+5HaXnPl0jAFukgY2VNeyaR/LOR/V23hlnPSuHfkadoVUYKmQBcJE9tKyrh52lyWbtzJI1f25obBnb0uSSJMUP/6zewiM8s2sxwzm3yYdgPNrMrMrgldiSL+99223Vz97Fdkby7mr2PTFeZyVBocoZtZFPA0MBzIA+aa2Qzn3Ip62j0GfNQYhYr41aLcHdw8bS4B5/jHD4fQv1N7r0uSCBXMCH0QkOOcW+ucKwcygVH1tLsTmA5sCWF9Ir722crNjJk6h9Yto5h++9kKczkmwcyhnwzk1lrOAwbXbmBmJwNXAhcAAw/1QGY2EZgIkJycTFZW1hGWG35KSkp80Y9gqb+hk5VbwYvLy+l8XAt+2tfYsHweGxrlmYKn1zeyBRPo9Z22zdVZ/jNwr3Ou6nBneXPOTQWmAqSnp7uMjIzgqgxjWVlZ+KEfwVJ/j51zjj99spppy3M4r/vxPHNDf+LD5GAhvb6RLZjfojygY63lVCC/Tpt0ILMmzJOAi82s0jn3TiiKFPEL5xwPzFjOS1+v59oBqfz2qjO0W6KETDCBPhfoZmZpwEZgNHB97QbOubR9t81sGvC+wlzkQFUBx31vLyVzbi4//F4a/3lxT523XEKqwUB3zlWa2SSq916JAl5wzi03s9tqtk9p5BpFIl4g4LjnzSVMX5DHnRd05WfDuyvMJeSCmrhzzs0EZtZZV2+QO+duPPayRPzDOcdD769g+oI87hrWjbuGdfe6JPEpTd6JNLLHP8pm2lffccs5afxkaDevyxEfU6CLNKLnv1zHs1lruGFwJ+67RHPm0rgU6CKNZNayAv7rgxVcdPqJPDSqt8JcGp0CXaQRzPuuiJ9kLqJfx3b8eXQ/oloozKXxKdBFQmzt1hJueWkeJ7VrxfPjBxIXows4S9NQoIuE0I495UyYNpcWZkybMJDE+FivS5JmRIEuEiJVAcdPMheRv2MvfxuXTucO8V6XJM1MeJxAQsQHfv9xNl+s3spvrzyDAZ111kRpehqhi4TArGUFPJu1hjGDOnH94E5elyPNlAJd5BityN/FT19bzJmd2vHAZb28LkeaMQW6yDEo2l3OxL/Po22rGP46doD2aBFPaQ5d5ChVVgWY9I8FbCku441bz+KEhDivS5JmTiN0kaP0+EfZfLWmkEeu6E3fju28LkdEgS5yNGYt28TU2WsZO6Qz16Z3bPgOIk1AgS5yhDYU7uHuNxfTN7Utv7q0p9fliOynQBc5AhVVAe7MXIgBf7m+Py2j9SGohA99KCpyBB75YCWLc3fw9PX96ZjY2utyRA6gEbpIkN5akLf/QhWX9EnxuhyRgyjQRYKQs6WE+95exqC0RH55sebNJTwp0EUaUFZZxU8yFxIX04KnRp+pc5tL2NIcukgDfjNjBcvzdzF17ABObKuDhyR8KdBFDuNfBZW8ungDt2ecyoWnn+h1OSKHpSkXkUPI3lTMc0vL6N+pHT8b3t3rckQapEAXqceu0gpue3k+rWOMKWMHEBOlPxUJf/otFakjEHD87LXF5Bbt4Y5+LXXSLYkYCnSROp7+PIdPV27mV5f0pHt7HQkqkUOBLlJLVvYW/vjpaq7odxLjz+7idTkiR0SBLlJj44693PXaInokJ/C7q/pgpv3NJbIo0EWA8srqi1VUVjme/cEAWsVqqkUij/ZDFwEen7WKhRuqT7qVlhTvdTkiRyWoEbqZXWRm2WaWY2aT69l+g5ktqfn6ysz6hr5Ukcbx8fJNPPflOsad1Vkn3ZKI1mCgm1kU8DQwEugFjDGzupc2Xwec55zrAzwMTA11oSKNIbdoD794YzFnnNyW+y7RSbcksgUzQh8E5Djn1jrnyoFMYFTtBs65r5xz22sW5wCpoS1TJPT2lldx69/n44CndbEK8YFg5tBPBnJrLecBgw/T/mbgw/o2mNlEYCJAcnIyWVlZwVUZxkpKSnzRj2D5pb/OOf62tJyVBZXcNaAla5d+w9p62vmlv8FSfyNbMIFe375brt6GZudTHejn1LfdOTeVmumY9PR0l5GREVyVYSwrKws/9CNYfunvC1+u46v8Ffx0WHd+MqzbIdv5pb/BUn8jWzCBngfUvqx5KpBft5GZ9QGeA0Y65wpDU55I6H29ppDfzlzJsJ7J3HlBV6/LEQmZYObQ5wLdzCzNzGKB0cCM2g3MrBPwFjDWObc69GWKhMb6wt3c9vJ8uiTF84fv96WFLlYhPtLgCN05V2lmk4CPgCjgBefccjO7rWb7FOB+oAPwTM3RdZXOufTGK1vkyJWUVfLDl+YB8ML4gbRtFeNxRSKhFdSBRc65mcDMOuum1Lp9C3BLaEsTCZ1AwHFX5iLWbN3NixMG0alDa69LEgk5HfovzcIfP1nNpys3c9/FPTmnW5LX5Yg0CgW6+N4b83L5y+c5jB7YkQn/0cXrckQajQJdfO2rNdv45VtLOadrEg9f0VtnUBRfU6CLb+VsKeG2v88nLSmeZ37QX5eRE9/Tb7j4UmFJGTdNm0tsdAteuHEgx8VpjxbxP50+V3ynuLSCG/9nLpt3lZI5cQgdE7VHizQPGqGLr5RWVPHDl+axsmAXz/6gP2d2au91SSJNRiN08Y2KquqrDs1ZW8Sfr+vHBacle12SSJPSCF18oSrg+Olri/h05RYevqI3V5x5stcliTQ5BbpEvEDAMXn6Et5fUsAvR57G2CGdvS5JxBMKdIlozjkenbWKN+bn8eOh3bj1vFO9LknEMwp0iVjOOX7/cTZTZ69l7JDO/PQw5zUXaQ70oahEJOccT3yUzTNZaxgzqBMPXn66jgKVZk+BLhEnEHA8+N5yXvx6PWMGdeKRK3rrvOYiKNAlwpRXBpj81hLeWrCRH34vjf+8uKdG5iI1FOgSMbbvLue2l+fzr3VF/Gx4d+68oKvCXKQWBbpEhDVbS7hp2lwKdpTy5+v6aT9zkXoo0CXsvb0wj/veXkZcTBSvThzCgM46nF+kPgp0CVt7y6v4zYzlvDYvl0FdEnlqzJmc2DbO67JEwpYCXcJSzpZi7nhlIau3FDPp/K7cNawb0TqfuchhKdAlrFQFHC99/R2PzVpF69hopk0YxHndj/e6LJGIoECXsJG/Yy+/eGMxX60pJKPH8Tx+dR9OOE5TLCLBUqCL58orA7zwf+t46rNvcQ4ev7oP16anapdEkSOkQBfPBAKOmcsK+OMnq1m7dTfDep7AA5edrisMiRwlBbo0uaqAY+bSAv7yvzlkby6m6wlteOHGdF2QQuQYKdClyQQCjk9WbuZPn6xm1abqIH9ydD8u7XMSUToXi8gxU6BLoyuvDPD2wjymzl7Lmq276dKhNU+NOZNLzkhRkIuEkAJdGs2qTbt4c14e7yzayLaScnqlHMeTo/txyRkp2qdcpBEo0CWk1m3bzXuL85m5tIBVm4qJiTKGnpbM9YM78b1uSdpzRaQRKdDlmOzYU84364qY+10RWdlb+XZLCWYwoFN7fnNZLy7vdzKJ8bFelynSLCjQJWilFVWs3VlF/r82sCx/J/O/20725mIAYqNaMKBze8YM6sSI3idycrtWHlcr0vwEFehmdhHwJBAFPOece7TOdqvZfjGwB7jRObcgxLVKI6sKOHbsKadwdzmFJeXk79jL+qI9rNu2m1UFu1iztYSAA1hKQsto+nVqx2V9UxjYJZG+HdsRFxPldRdEmrUGA93MooCngeFAHjDXzGY451bUajYS6FbzNRh4tua7NJLKqgDlVQHKK6u/yioDVNRZV14ZoKxmuaLme0lZJYUl5RTuLqOoJriLdleH+PY95Th34PO0MDipXSt6JCcwsveJBIpyue7Cs0lt30rz4SJhJpgR+iAgxzm3FsDMMoFRQO1AHwW85JxzwBwza2dmKc65glAX/MXqrTz8/gpcTfIckD/ugG8HtXH7t7sDl+uE2JHct6y8nNj/+7TWYxy4Pah66valTj9qt6lyjvLKQM1I+eiYQbtWMSTGx9IhviVdT2jDoPhYOsTHkhgfS2KbliTFx5LcNo6O7VsTG/3vPVKysgp0JKdImAom0E8Gcmst53Hw6Lu+NicDBwS6mU0EJgIkJyeTlZV1hOVCzvYq2reoqH68erbvW1d38Gh1bljNjbqPUft+h9u2b3tFRYDYmKo6j3245z2yeq32GgOjBTEtoohuATEtILqFHXT7UNtiWkDLKKNNDLX2/y6r+apRAWyH8u2Qm3fgiwpQUlJyVK9bpFJ//c1v/Q0m0OvLzbrjw2Da4JybCkwFSE9PdxkZGUE8/YEygFuO+F6NJysri6PpR6RSf/1N/Y1swRzdkQd0rLWcCuQfRRsREWlEwQT6XKCbmaWZWSwwGphRp80MYJxVGwLsbIz5cxERObQGp1ycc5VmNgn4iOrdFl9wzi03s9tqtk8BZlK9y2IO1bstTmi8kkVEpD5B7YfunJtJdWjXXjel1m0H3BHa0kRE5EjoDEkiIj6hQBcR8QkFuoiITyjQRUR8wlx9x703xRObbQXWe/LkoZUEbPO6iCak/vqb+hv+Ojvnjq9vg2eB7hdmNs85l+51HU1F/fU39TeyacpFRMQnFOgiIj6hQD92U70uoImpv/6m/kYwzaGLiPiERugiIj6hQBcR8QkF+lEws4fNbImZLTKzj83spFrbfmlmOWaWbWYjvKwzVMzsCTNbVdPnt82sXa1tfuzvtWa23MwCZpZeZ5vv+gvVF4Kv6VOOmU32up7GYGYvmNkWM1tWa12imX1iZt/WfG/vZY3HSoF+dJ5wzvVxzvUD3gfuBzCzXlSfL/504CLgmZqLbEe6T4Dezrk+wGrgl+Dr/i4DrgJm117p1/7WuhD8SKAXMKamr34zjerXrbbJwGfOuW7AZzXLEUuBfhScc7tqLcbz78vtjQIynXNlzrl1VJ8fflBT1xdqzrmPnXOVNYtzqL4iFfi3vyudc9n1bPJlf6l1IXjnXDmw70LwvuKcmw0U1Vk9Cnix5vaLwBVNWVOoKdCPkpk9Yma5wA3UjNA59MWy/eQm4MOa282hv7X5tb9+7VcwkvddXa3m+wke13NMgrrARXNkZp8CJ9az6T7n3LvOufuA+8zsl8Ak4AGCvFh2OGqovzVt7gMqgVf23a2e9r7pb313q2ddRPS3AX7tV7OjQD8E59ywIJv+A/iA6kCP2ItlN9RfMxsPXAoMdf8+eMG3/T2EiO1vA/zar2BsNrMU51yBmaUAW7wu6FhoyuUomFm3WouXA6tqbs8ARptZSzNLA7oB3zR1faFmZhcB9wKXO+f21Nrky/4ehl/7G8yF4P1qBjC+5vZ44FDvziKCRuhH51Ez6wEEqD4F8L4LZi83s9eBFVRPTdzhnKvyrsyQ+QvQEvjEzADmOOdu82t/zexK4L+B44EPzGyRc26EX/t7qAvBe1xWyJnZq0AGkGRmeVS/q34UeN3MbgY2ANd6V+Gx06H/IiI+oSkXERGfUKCLiPiEAl1ExCcU6CIiPqFAFxHxCQW6iIhPKNBFRHzi/wHuo16KaHhhEQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_CDF(s)\n",
    "grid()\n",
    "legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "633ae908",
   "metadata": {},
   "source": [
    "### Are we stuck?\n",
    "\n",
    "* **No!** we can approximate the CDF using a sample\n",
    "* Estimating the CDF does not require many examples (proof: more advanced probability)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "e55cfea8",
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.random import choice"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "9b7e2926",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x15c20e4c0>"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#figure(figsize=[15,10])\n",
    "for m in [50,500,80000]:\n",
    "    sample=choice(s,m)\n",
    "    plot_CDF(sample)\n",
    "grid()\n",
    "legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3efe6a4b",
   "metadata": {},
   "source": [
    "###  Problem in HW5 \n",
    "\n",
    "Write a program that sorts `S` in linear time using bucket sort.\n",
    "\n",
    "The steps for the algorithm are:\n",
    "1. Sample 1000 points from `S`\n",
    "2. Compute 99 pivot points at the 1%,2%, ,99% of the distribution\n",
    "3. Use the pivot points to define 100 buckets.\n",
    "4. collect into each bucket the elements of `S` that are between i%      and (i+1)%\n",
    "5. Compute the sizes of the least filled and the most filled bucket. How do these sizes impact the running time?\n",
    "6. use python sort on each bucket.\n",
    "7. concatanate the buckets to form the sorted list.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6296a938",
   "metadata": {},
   "source": [
    "### Submitting the homework\n",
    "\n",
    "You can use this notebook to test your code. However, to submit the code write it out into the homework and submit through grade-scope."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4fb1e529",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}