{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": true, "editable": true, "inputHidden": false, "outputHidden": false }, "source": [ "# basics of probability theory\n", "\n", "In the context of a [course in Computational Neuroscience](http://invibe.net/LaurentPerrinet/Presentations/2017-03-06_cours-NeuroComp), I am teaching a basic introduction in [Probabilities, Bayes and the Free-energy principle](http://blog.invibe.net/files/2017-03-13_NeuroComp_FEP.html).\n", "\n", "Let's learn to use probabilities in practice by generating some \"synthetic data\", that is by using the computer's number generator. \n", "\n", "\n", "\n", "\n", "\n", "\n", "Let's begin with a dice:\n", "\n", "\n", "## dice" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "np" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.logistic" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Help on built-in function randint:\n", "\n", "randint(...) method of mtrand.RandomState instance\n", " randint(low, high=None, size=None, dtype='l')\n", " \n", " Return random integers from low (inclusive) to high (exclusive).\n", " \n", " Return random integers from the \"discrete uniform\" distribution of\n", " the specified dtype in the \"half-open\" interval [low, high). If\n", " high is None (the default), then results are from [0, low).\n", " \n", " Parameters\n", " ----------\n", " low : int\n", " Lowest (signed) integer to be drawn from the distribution (unless\n", " high=None, in which case this parameter is one above the\n", " *highest* such integer).\n", " high : int, optional\n", " If provided, one above the largest (signed) integer to be drawn\n", " from the distribution (see above for behavior if high=None).\n", " size : int or tuple of ints, optional\n", " Output shape. If the given shape is, e.g., (m, n, k), then\n", " m * n * k samples are drawn. Default is None, in which case a\n", " single value is returned.\n", " dtype : dtype, optional\n", " Desired dtype of the result. All dtypes are determined by their\n", " name, i.e., 'int64', 'int', etc, so byteorder is not available\n", " and a specific precision may have different C types depending\n", " on the platform. The default value is 'np.int'.\n", " \n", " .. versionadded:: 1.11.0\n", " \n", " Returns\n", " -------\n", " out : int or ndarray of ints\n", " size-shaped array of random integers from the appropriate\n", " distribution, or a single such random int if size not provided.\n", " \n", " See Also\n", " --------\n", " random.random_integers : similar to randint, only for the closed\n", " interval [low, high], and 1 is the lowest value if high is\n", " omitted. In particular, this other one is the one to use to generate\n", " uniformly distributed discrete non-integers.\n", " \n", " Examples\n", " --------\n", " >>> np.random.randint(2, size=10)\n", " array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0])\n", " >>> np.random.randint(1, size=10)\n", " array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])\n", " \n", " Generate a 2 x 4 array of ints between 0 and 4, inclusive:\n", " \n", " >>> np.random.randint(5, size=(2, 4))\n", " array([[4, 0, 2, 1],\n", " [3, 2, 2, 0]])\n", "\n" ] } ], "source": [ "help(np.random.randint)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "array([5, 4, 2, 5, 3, 5, 5, 0, 4, 1])" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.randint(6, size=10)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": true, "editable": true, "inputHidden": false, "outputHidden": false }, "source": [ "## a note on RNGs\n", "\n", "On a computer, randomness is (possibly) deterministic !\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "array([3, 3, 0, 3, 4, 4, 3, 1, 4, 3])" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.randint(6, size=10)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "array([3, 4, 2, 4, 4, 1, 2, 2, 2, 4])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.seed(42)\n", "np.random.randint(6, size=10)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "array([4, 0, 1, 5, 2, 0, 3, 1, 3, 3])" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.seed(43)\n", "np.random.randint(6, size=10)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "array([3, 4, 2, 4, 4, 1, 2, 2, 2, 4])" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.seed(42)\n", "np.random.randint(6, size=10)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "data": { "text/plain": [ "array([1, 5, 2, 1, 0, 4, 0, 2, 1, 1])" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.random.seed(None)\n", "np.random.randint(6, size=10)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": true, "editable": true, "inputHidden": false, "outputHidden": false }, "source": [ "## coin" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1 1 0 1 1 0 1 1 0 1]\n" ] } ], "source": [ "result = np.random.randint(2, size=10)\n", "print(result)" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "(0.69999999999999996, 0.45825756949558394, 0.20999999999999996)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "result.mean(), result.std(), result.var()" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean 0.495 , std 0.499974999375\n" ] } ], "source": [ "N = 1000\n", "result = np.random.randint(2, size=N)\n", "print('Mean ', result.mean(), ', std ', result.std())" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean [ 0.54 0.516 0.475 0.501 0.498 0.505 0.488 0.499 0.495 0.498\n", " 0.512 0.503 0.515 0.512 0.501 0.504 0.491 0.465 0.475 0.49\n", " 0.496 0.508 0.508 0.498 0.51 0.49 0.519 0.495 0.503 0.521\n", " 0.513 0.492 0.455 0.474 0.509 0.507 0.49 0.489 0.494 0.493\n", " 0.509 0.497 0.494 0.495 0.482 0.514 0.493 0.494 0.501 0.517\n", " 0.492 0.525 0.507 0.501 0.483 0.501 0.497 0.469 0.49 0.515 0.5\n", " 0.487 0.49 0.488 0.502 0.499 0.495 0.49 0.538 0.5 0.499\n", " 0.513 0.508 0.5 0.491 0.492 0.516 0.481 0.503 0.518 0.514\n", " 0.487 0.497 0.489 0.489 0.523 0.498 0.491 0.495 0.526 0.5 0.513\n", " 0.482 0.507 0.488 0.489 0.487 0.501 0.485 0.509] , std 0.0140053239877\n" ] } ], "source": [ "N = 1000\n", "N_trials = 100\n", "result = np.random.randint(2, size=(N, N_trials))\n", "print('Mean ', result.mean(axis=0), ', std ', result.mean(axis=0).std())" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Grand average= 0.49903\n" ] } ], "source": [ "print('Grand average=', result.mean())" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": true, "deletable": true, "editable": true, "outputExpanded": false }, "outputs": [], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean [ 0.511 0.52 0.528 0.508 0.488 0.516 0.498 0.501 0.495 0.489\n", " 0.507 0.486 0.509 0.503 0.539 0.515 0.519 0.5 0.495 0.505\n", " 0.489 0.501 0.512 0.507 0.479 0.502 0.525 0.489 0.514 0.498\n", " 0.494 0.533 0.506 0.518 0.519 0.509 0.486 0.51 0.483 0.52\n", " 0.496 0.482 0.503 0.495 0.456 0.526 0.499 0.464 0.539 0.521\n", " 0.482 0.505 0.477 0.521 0.518 0.499 0.51 0.511 0.5 0.494\n", " 0.496 0.502 0.497 0.486 0.51 0.486 0.536 0.493 0.497 0.512\n", " 0.527 0.512 0.492 0.53 0.487 0.479 0.493 0.5 0.521 0.478\n", " 0.51 0.475 0.522 0.498 0.508 0.486 0.487 0.511 0.52 0.49\n", " 0.498 0.504 0.495 0.499 0.503 0.508 0.499 0.502 0.477 0.5 ] , std 0.0157819517171\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 1000\n", "N_trials = 100\n", "result = np.random.randint(2, size=(N, N_trials))\n", "print('Mean ', result.mean(axis=0), ', std ', result.mean(axis=0).std())\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=np.linspace(0, 1, 100));" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean [ 0.479 0.503 0.497 0.504 0.498 0.501 0.529 0.495 0.523 0.496\n", " 0.527 0.486 0.495 0.506 0.498 0.53 0.509 0.487 0.483 0.511\n", " 0.485 0.526 0.491 0.483 0.496 0.477 0.515 0.519 0.509 0.517\n", " 0.505 0.497 0.515 0.517 0.506 0.504 0.491 0.482 0.528 0.5 0.526\n", " 0.517 0.485 0.497 0.495 0.518 0.482 0.502 0.489 0.499 0.512\n", " 0.511 0.475 0.525 0.501 0.486 0.496 0.492 0.466 0.474 0.515\n", " 0.507 0.477 0.504 0.516 0.52 0.48 0.511 0.48 0.482 0.486\n", " 0.493 0.501 0.535 0.466 0.517 0.505 0.499 0.529 0.499 0.49\n", " 0.501 0.503 0.497 0.521 0.499 0.511 0.5 0.474 0.482 0.48\n", " 0.508 0.509 0.498 0.489 0.515 0.516 0.516 0.486 0.524] , std 0.0157702853494\n" ] }, { "data": { "image/png": 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LVAAAwMqOHPvd/ViSb1/jLAAAwBp5600AABhK7AMAwFBiHwAAhhL7AAAwlNgHAIChxD4A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\nAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 1000\n", "N_trials = 100\n", "result = np.random.randint(2, size=(N, N_trials))\n", "print('Mean ', result.mean(axis=0), ', std ', result.mean(axis=0).std())\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=np.linspace(0, 1, 100));" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean [ 0.5011 0.5041 0.5005 0.5067 0.5032 0.4995 0.4971 0.4955 0.5015\n", " 0.5007 0.494 0.5011 0.5096 0.4999 0.5026 0.4957 0.501 0.5026\n", " 0.4958 0.5032 0.4963 0.5 0.5026 0.5034 0.4952 0.5011 0.4997\n", " 0.5093 0.4984 0.5014 0.4954 0.4989 0.4913 0.5009 0.5028 0.493\n", " 0.4945 0.4978 0.4915 0.5052 0.4921 0.5 0.5025 0.4997 0.4981\n", " 0.4974 0.5049 0.4991 0.5078 0.5018 0.4962 0.4976 0.498 0.5034\n", " 0.5003 0.4972 0.5028 0.5005 0.5024 0.4973 0.4928 0.5032 0.4942\n", " 0.499 0.5004 0.4912 0.5016 0.5043 0.4966 0.5007 0.4991 0.5002\n", " 0.4989 0.5114 0.4995 0.5002 0.5009 0.5028 0.5035 0.4974 0.5052\n", " 0.4942 0.498 0.5001 0.5051 0.4962 0.4843 0.499 0.496 0.4992\n", " 0.5033 0.4913 0.5043 0.4986 0.5018 0.4934 0.4963 0.5063 0.5065\n", " 0.5078] , std 0.00448078118189\n" ] }, { "data": { "image/png": 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GEvoAADCQ0AcAgIGEPgAADCT0AQBgIKEPAAADCX0AABhI6AMAwEBCHwAABhL6AAAwkNAH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\nSOgDAMBAQh8AAAb6v6t3FWuz1zqMAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 10000\n", "N_trials = 100\n", "result = np.random.randint(2, size=(N, N_trials))\n", "print('Mean ', result.mean(axis=0), ', std ', result.mean(axis=0).std())\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=np.linspace(0, 1, 100));" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean [ 0.56 0.5 0.52 0.48 0.42 0.5 0.5 0.53 0.47 0.46 0.51 0.46\n", " 0.46 0.45 0.52 0.47 0.52 0.41 0.43 0.49 0.51 0.58 0.47 0.47\n", " 0.57 0.44 0.44 0.52 0.56 0.46 0.55 0.54 0.47 0.52 0.51 0.57\n", " 0.43 0.56 0.51 0.51 0.58 0.54 0.44 0.44 0.55 0.47 0.52 0.47\n", " 0.52 0.47 0.57 0.52 0.46 0.47 0.58 0.5 0.48 0.47 0.42 0.5\n", " 0.52 0.51 0.49 0.49 0.53 0.45 0.53 0.5 0.54 0.52 0.56 0.49\n", " 0.54 0.53 0.42 0.45 0.5 0.4 0.43 0.55 0.56 0.49 0.46 0.46\n", " 0.5 0.61 0.52 0.57 0.51 0.56 0.56 0.53 0.49 0.51 0.4 0.56\n", " 0.51 0.51 0.49 0.45] , std 0.0459995652153\n" ] }, { "data": { "image/png": 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3VtXhVQ0FAAAsZ8ehX1XnJHlDkhcmuSTJ1VV1yaoGAwAAdm6ZI/rPTnJnd9/V3Q8meUeS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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 100\n", "N_trials = 100\n", "result = np.random.randint(2, size=(N, N_trials))\n", "print('Mean ', result.mean(axis=0), ', std ', result.mean(axis=0).std())\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=np.linspace(0, 1, 100));" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false, "deletable": true, "editable": true, "inputHidden": false, "outputHidden": false }, "source": [ "## normal variable" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false, "deletable": true, "editable": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Mean [ 0.50195 0.49787 0.50032 ..., 0.50307 0.49871 0.49993] , std 0.00160510109458\n" ] }, { "data": { "image/png": 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EAABAM0IAAAA0IwQAAEAzQgAAADQjBAAAQDN7Ft0AgM1aOnxs0U0AgB1FCACAObhQOD15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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 100000\n", "N_trials = 10000\n", "result = np.random.randint(2, size=(N, N_trials))\n", "print('Mean ', result.mean(axis=0), ', std ', result.mean(axis=0).std())\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=np.linspace(0.49, .51, 100));" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "Central limit theorem:\n", "\n", "Over multiple measurements, the average of an identical random variable converges to a normal law :\n", "$$\n", "p (x) = \\frac {1} { \\sqrt{ 2\\pi } \\sigma} \\cdot \\exp{(- \\frac {1} {2} \\cdot \\frac {(x - m)^2} {\\sigma^2} )}\n", "$$\n", "\n", "(moreover, we know from this theorem that the asymptotic mean is the mean or the rv and that the variance decreases inversely proportionally with the number of measurements)" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "data": { "image/png": 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L+wyH4mJYutRDwyTlPEuAJJ2kLgf3MWxXJYeK2rCi15DQcdRMqovawKRJye5A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\nCqQsc801yeP06VBdHTaLpLz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G7NzIEz/7P3x85czQsSQ1gSVAUr1rl73ELx6ZRq8D7/La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\nzMysYf4PvNcWEIdm0W8AAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mean = result.mean(axis=0).mean()\n", "std = result.mean(axis=0).std()\n", "bins = np.linspace(0.49, .51, 100)\n", "\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=bins, normed=True)\n", "ax.plot(bins, 1 / np.sqrt(2*np.pi) / std * np.exp(-.5*(bins-mean)**2/std**2), 'r', lw=2 );" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "The central limit theorem works with all forms of random variables:" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[ 0.44869862 0.57662481 0.18603079 ..., 0.15279605 0.59446338\n", " 0.57548346]\n", " [ 0.17774894 0.49783666 0.27971604 ..., 0.62650323 0.43497046\n", " 0.14628423]\n", " [ 0.55741204 0.06820418 0.68217069 ..., 0.69438983 0.56086557\n", " 0.0021194 ]\n", " ..., \n", " [ 0.29663497 0.24120372 0.57932703 ..., 0.89435831 0.5468314\n", " 0.07252068]\n", " [ 0.88196308 0.37744147 0.2177457 ..., 0.02152461 0.77621035\n", " 0.50561815]\n", " [ 0.61974228 0.08479278 0.48558868 ..., 0.91904245 0.18800096\n", " 0.63077166]]\n" ] } ], "source": [ "N = 100000\n", "N_trials = 1000\n", "result = np.random.rand(N, N_trials)\n", "print(result)" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "data": { "image/png": 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U/t3sGjiUzcNGpy7nRUohYNY2WwIkKSVDgCQ1kKMGBdfQzEAlR3UHijFxNZJUXIYASWog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\nknSsHTvgC1/I9v/8z9PWopq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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mean = result.mean(axis=0).mean()\n", "std = result.mean(axis=0).std()\n", "bins = np.linspace(0.49, .51, 100)\n", "\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=bins, normed=True)\n", "ax.plot(bins, 1 / np.sqrt(2*np.pi) / std * np.exp(-.5*(bins-mean)**2/std**2), 'r', lw=2 );" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "collapsed": false, "deletable": true, "editable": true, "scrolled": false }, "outputs": [ { "data": { "image/png": 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GdtwRfvghXS1IkiLIiR1AkgCYMAFuuAGAK/Y+HUKIHEgqvfyCEcs9P/H666Ft\nWxg0CE4/HfLzV/6cfu3LNkMJX09S9eBIgKTKoUcPWLKEx7bch0+abhQ7jVS+dtwRjj0WFi1yyVBJ\nUVgCJMU3ahQ89RTUqcOA3TvFTiNVjH79oHZtePRReOut2GkkZRhLgKS4Cguha9f0uKCAGXUbxs0j\nVZSWLaF79/T4wguhqChuHkkZxRIgKa6hQ+Gjj9IfiC66KHYaqWL16AHNmsGYMfDAA7HTSMoglgBJ\n8cyZAxdfnB73759eGiFlkjp1fl8hqFcvmDs3bh5JGcMSICmefv3gxx/TSZJHHx07jRTHccdBmzbw\n/fcwYEDsNJIyRKlKQAihSwjh0xDCuBDCQyGEWiGE9UIIo0MIE0IIj4QQapZVWEnVyHffpcsjAlx/\nvUuCKnNlZaX/BgAGDqTZ7Olx80jKCCUuASGE5kBnoHWSJFsC2cDRQH/g+iRJNgR+AU4ti6CSqpmC\ngnR5xGOPTUcCpEy2885w1FGwcCE9Xx0aO42kDFDay4FygNohhBxgDWAasBfwePH9Q4FDSvkekqqb\nt9+Ghx+GWrXgmmtip5Eqh/79oVYtOox/jW2+/yJ2GknVXIlLQJIkU4FrgUmkP/z/CowFZiVJsrT4\nYVOA5st7fgjhjBDCmBDCmBkzZpQ0hqSqpqgIunRJj7t1g3XWiZtHqizWXfe/y+X2HnkXJEnkQJKq\ns9JcDrQW0AFYD2gG1AH2X9XnJ0kyOEmS1kmStM7LyytpDElVzUMPwXvvQdOm7pQq/VFBATPqNGC7\n7z/noM/fiJ1GUjWWU4rn7gN8myTJDIAQwpPAzkCDEEJO8WhAC2Bq6WNKKm/5BSNWeN/Efu3L5k3m\nz0/nAgD07Qt165bN60rVRb16XLvrCfR/4WYKXr2PlzbakUU5rq8hqeyVZk7AJGDHEMIaIYQA7A18\nBowCDi9+zInA06WLKKnaGDQIpkyBVq2gU6fYaaRK6bGt9mF8Xj4tZk/n1PeHx44jqZoqzZyA0aQT\ngD8APil+rcFAT6BrCGEC0AgYUgY5JVV133//+6ZI112XLoso6U+KsrK5cq/TADjn3cdoPO+XyIkk\nVUel+r9wkiSXJUmyaZIkWyZJckKSJIuSJPkmSZLtkyTZMEmSI5IkWVRWYSVVYZdeml4O1LEj7L57\n7DRSpfZ2/ja8tOH21F28gK5vPBA7jqRqyK/iJJW/Dz6A++6DGjXcEVVaRdfscQpLsrI56uOX2HT6\nt7HjSKroaMk6AAAgAElEQVRmLAGSyleSwEUXpb+ffz5suGHsRFKV8E2jFjyw7YFkJ0Vc8soQlwyV\nVKYsAZLK19NPw6uvQqNG0Lt37DRSlXLDzscyq1Zddv3uIxix4hW8JGl1WQIklZ/Fi6F79/T48suh\nQYOocaSq5tfa9bhpp2PSG926wZIlcQNJqjYsAZLKz623woQJsOmmcOaZsdNIVdL9rQ7km7WawRdf\nwB13xI4jqZqwBEgqHz//DFdckR5fe206KVjSaluSXYO+e56a3rj8cvjFJUMllZ4lQFL56NMHZs2C\ndu3gwANjp5GqtJc33B723BNmzoQrr4wdR1I1YAmQVPY+/xxuuy3dEGzQIAghdiKpagsh3WQvBLjl\nFvjqq9iJJFVxObEDSKqGuneHwkI44wzYaqvYaaRylV9QQav2bLMNnHwy3HNP+m9s+PCKzyCp2nAk\nQFLZevllePZZqFfv9zkBksrGVVdBnTrp0rujRsVOI6kKswRIKjuFhdC1a3p88cXQpEncPFJ107Qp\n9OqVHnfpkv6bk6QSsARIKjtDhsAnn0B+Plx4Yew0UvXUtSu0bAn/938wdGjsNJKqKEuApLIxe/bv\nOwL37w+1asXNI1VXtWtDv37p8SWXwJw5cfNIqpIsAZLKRt++MH067LwzHHFE7DRS9XbMMbDDDvDD\nD2nplqTVZAmQVHrffgvXX58eX3+9S4JK5e23JUMBBg2i2ezpcfNIqnIsAZJKr2dPWLwYjj8e2rSJ\nnUbKDDvtBEcdBQsX0uM15wZIWj2WAEml8+ab8Nhj6XXKffvGTiNlln79IDeXQz57jW2nfh47jaQq\nxBIgqeSKitJlCgG6dUtXLJFUcfLz/7ss7z9H3gVJEjePpCrDEiCp5P71LxgzJl27vEeP2GmkzNSr\nFzPqNGDbaV9w8PjXYqeRVEVYAiSVzPz5v29a1Lcv1K0bN4+UqerVY+CunQAoePU+ai1ZGDmQpKog\nJ3YASWUvv2DECu+b2K992bzJwIEwdSq0agWdOpXNa0oZ4q/+jZbE41vtzYkfPMsW07/h9Pee4uad\njynT15dU/TgSIGn1TZ78+9rk118PWX6USDEVZWVz5d6nAXD26MdpMuenyIkkVXb+n1vS6uvVCxYs\ngMMPh912i51GEvDuOlvz/MY7scaSRfR4fVjsOJIqOUuApNXz7rvphODcXBgwIHYaScu4Zo+TWZSd\nw2HjXmHraV/GjiOpErMESFp1RUVw4YXpcdeusN56cfNI+h+T1mrKva07ANB75N0uGSpphSwBklbd\ngw/C6NHwt7/9vjKQpErllrZHMWONBrSZ+hntP38zdhxJlZQlQNKqmTcPCgrS4759oV69uHkkLdfc\n3DUYtOvxAPR69V5ylyyKnEhSZWQJkLRqBgz4fUnQE0+MnUbSX3h063aMz8unxezpnPb+8NhxJFVC\nlgBJKzdp0u+TgG+4wSVBpUquKCubK/Y+HYBz3n2Mtef8HDmRpMrG/5NLWrmCAli4EI48EnbdNXYa\nSavgnXX/zgsbt6XOkoX0fH1o7DiSKhlLgKS/1GrKeHjooXRJ0N82CJNUJVy956n/XTJ0m++/iB1H\nUiViCZC0QiEp4p+vDE5vdOsG+flR80haPZMb/I2723QE4PKX7yQkRZETSaoscmIHkDJFfsGIFd43\nsV/7Ckyy6g4b9wrbTPuKH+s2ZM/52zD/L/4Mf/RXf6a/+ruQVLZu2/EIDh83km2mfUnHT0fx5JZ7\nx44kqRJwJEDSctVdNJ+er90HQL/dT2J+zdpxA0kqkXm5a9B/93RFr56vDaXOovmRE0mqDCwBkpbr\nvLcfJm/eLD5otgnDt9gjdhxJpfDUFnvyUdONaTJ3Jue8+1jsOJIqAUuApD9Zb+ZUThnzDACX73Mm\nSfCjQqrKkpBFn73PAOC095+i5awfIieSFJv/Z5f0J5e+cjc1i5by6Fb78HHTjWPHkVQGPmy+KU9s\nsSe5hUu5ZNSQ2HEkRWYJkPQ/9vj6ffb++n3m1KzNwN3cGViqTvrvfhLzatRi/y/fgZEjY8eRFJEl\nQNJ/1ShcQu9X7gbgxp2PYUbdtSInklSWptdrxK1tj0xvXHghLF0aN5CkaCwBkv7rpDH/ZoOZU/m6\nYQuGbveP2HEklYMhbQ5h0ppNYNw4uOOO2HEkRVKqEhBCaBBCeDyE8HkIYXwIoW0IoWEI4aUQwlfF\nv/tVolQF5M39hc5vPwTAlXudxpLsGpETSSoPi3JqcvVep6Y3eveGGTPiBpIURWlHAm4EXkiSZFPg\n78B4oAAYmSTJRsDI4tuSKrnurw+l3uIFjNygDa9u0Dp2HEnl6D8btYV994VZs+CSS2LHkRRBiUtA\nCGFNYDdgCECSJIuTJJkFdACGFj9sKHBIaUNKKmfvv8+Rn7zM4qwcrtzrtNhpJJW3EODGGyEnB+6+\nG8aMiZ1IUgUrzUjAesAM4N4QwochhLtDCHWAJkmSTCt+zA9Ak+U9OYRwRghhTAhhzAyHIqV4iorg\n/PMBuKf1wUxs2DxyIEkVYtNNoUsXSBI477z0s0BSxihNCcgBWgG3J0myLTCPP1z6kyRJAiTLe3KS\nJIOTJGmdJEnrvLy8UsSQVCr33QejR/Nj3YbcstPRsdNIqki9e0PTpjB6NAwduvLHS6o2ckrx3CnA\nlCRJRhfffpy0BPwYQmiaJMm0EEJTYHppQ0oqJ7/8Aj17AnD1nqcwN3eNMnvp/IIRZfZakspJvXow\ncCAcfzwUFEDHjtCgQexUkipAiUcCkiT5AZgcQtik+NTewGfAM8BvOwydCDxdqoSSyk/v3vDTT7Db\nbjyz2e6x00iK4dhjYZddYPp0uPzy2GkkVZDSrg50PvCvEMLHwDZAX6Af0C6E8BWwT/FtSZXNRx/B\n7bdDdjbccks6UVBS5gkBbr4ZsrLSz4Jx42InklQBSlUCkiT5qPi6/q2TJDkkSZJfkiT5OUmSvZMk\n2ShJkn2SJJlZVmEllZEkgXPPTScCnncebLVV7ESSYtpmGzjrLCgshM6d088ISdWaOwZLmej+++Ht\nt6FJE+jTJ3YaSZXBlVdCo0YwahQ89ljsNJLKmSVAyjS//go9eqTHAwbAmmvGzSOpcmjYEPr2TY8v\nugjmzo2bR1K5sgRImeayy+DHH2HnneGEE2KnkVSZnHoqbLcdTJmSjgxIqrYsAVIG2WTGxHTi328T\nAJ0MLGlZ2dlw223pZ8N118Gnn8ZOJKmcWAKkTJEkXPHSHenEv7PPTicCStIfbb89nHkmLF2aLiDg\nJGGpWrIESBnikM9eZYfJ4yAvz2F+SX+tb9/0s+K11+Bf/4qdRlI5sARIGaD+wrlc8sqQ9Eb//rDW\nWnEDSarc1lor3UkY0knCs2bFzSOpzFkCpAzQ47Wh5M2fxegWW8BJJ8WOI6kq6NQJdt013Un40ktj\np5FUxnJiB5BUvrb5/guO/egFlmRlc+m+5/CSk4GlKim/YETFvmEI6SThbbZJfz/55HTloBL4q+wT\n+7UvaUJJpeBIgFSNZRcVcvV/biWLhLvbdOSrvHVjR5JUlWy5JXTpkk4OPvvsdGEBSdWCJUCqxk4c\n+yxbTP+GKfXX5qadjo4dR1JVdNll0Lw5vP8+3HVX7DSSyoglQKqm/jb7J7q++QAAl7U7kwU1a0VO\nJKlKqlsXbrghPe7VK50jIKnKswRI1VTvV+6i7uIF/GejHRm54Q6x40iqyg47DPbbL10lqHv32Gkk\nlQFLgFQN7fH1+7T/4i3m1ahFn33OiB1HUlUXQrrLeG4uDBsGI0fGTiSplCwBUnUzf366MzBww87H\n8n39tSMHklQtbLgh9O6dHp91FixYEDePpFKxBEjVTd++rPPrj4zPy+fe1gfHTiOpOuneHbbYAiZM\ngKuuip1GUilYAqTq5NNPYcAAAC7d9xyWZrsViKQyVLMmDB6cHg8YAOPGxc0jqcQsAVJ1UVgIp50G\nS5bw4N/3Z2yLzWMnklQd7bRTumfA0qVwxhlQVBQ7kaQSsARI1cXtt8O770LTplyz58mx00iqzq65\nBpo2hXfegTvvjJ1GUglYAqTqYPLkdP1ugFtvZU5unbh5JFVva64JN9+cHhcUwNSpcfNIWm1eMCyV\nsfyCERX7hkkC55wDc+fCoYdCx44wuoIzSKpWVulzLMnlrg13oN2E0Ty/15Gc3fHi8g8mqcw4EiBV\ndY8+Cs8++7/fzElSeQuBf7Y7i7k1a3PAl2/T7qt3YyeStBosAVJVNnMmdO6cHg8YAM2axc0jKaNM\nq5/HtbueAECfl+6gzqL5kRNJWlWWAKkq69YNpk+H3XZLVwaSpAo2rFV7Pmq6Ec3m/ETP14bGjiNp\nFVkCpKpq5Ei4917IzU3X7c7yn7OkileUlU3B/p1ZkpVNpw9HsMOkT2JHkrQK/KlBqooWLIAzz0yP\ne/eGTTaJm0dSRvt87fW4te2RAPR//iZqLVkYOZGklbEESFXRZZfB11/DVltB9+6x00gSt7Y9kvF5\n+eTPmka31++PHUfSSlgCpKrm3Xdh0KD08p+774aaNWMnkiSWZNeg+4EXsjRkccqYZ2g1ZXzsSJL+\ngiVAqkoWLICTToKionRS8Pbbx04kSf817m8bMniHQ8kiYeDzN5K7dHHsSJJWwBIgVSX//Cd88QVs\nthn06RM7jST9yY07H8uEhi3YYOYULnzzwdhxJK2AJUCqKt555/fLgO67D2rVip1Ikv5kUU5Nehx4\nAUUEznjvSbae9mXsSJKWwxIgVQW/XQaUJOlEYC8DklSJfdB8M4a06UB2UsSA526k5tIlsSNJ+oOc\n2AEkQX7BiBXeN7Ff+3QZ0C+/hM03h8svX+lzJGllyvszZNCux7PPhNFs+tN3nPvOo1y/63GrlWNi\nv/blGU/KeI4ESJXd22/DddellwHde6+XAUmqEhbWqEXPAy4A4Jx3H2XLHyZETiRpWZYAqRLLXbII\nTj45vQyoRw8vA5JUpbzXckvu3e4f1Cgq5PpnB6WfaZIqBUuAVIl1e+P+P10GJElVSf/dT+Trhi3Y\n6OfJdH99WOw4kopZAqRKqvWUTzn1/achOztdDSg3N3YkSVptC2vUostBXVkasjhtzNO0/e7j2JEk\nYQmQKqV6i+Zxw78HkUUCPXtCmzaxI0lSiX3cdGNu2ekoAAY+dz31Fs2LnEiSJUCqhC5/+U5azJ7O\nJ002gMsuix1HkkrtlrZH8X9/24gWs2dw2cuDY8eRMp4lQKpkDvz8TQ4b9woLcnK58B/doGbN2JEk\nqdSWZufQ9aCuLMypyeHjRrLfl2/HjiRltFKXgBBCdgjhwxDCs8W31wshjA4hTAghPBJC8CcYaRU1\nmfMTff9zCwBX73kKXzdqGTmRJJWdrxu1pN/uJwHQ94VbaDzvl7iBpAxWFiMBFwDjl7ndH7g+SZIN\ngV+AU8vgPaRqLyRFDHzuRhosnMuo9bfjgW0PjB1Jksrc0O0O4s11/06jBbO55oWb0yWQJVW4UpWA\nEEILoD1wd/HtAOwFPF78kKHAIaV5DylTnDT23+w28UN+rl2fHgdcCCHEjiRJZS4JWXQ/8EJm59ah\n3YT3OPLjl2JHkjJSaUcCbgB6AEXFtxsBs5IkWVp8ewrQfHlPDCGcEUIYE0IYM2PGjFLGkKq2jWdM\npODV+wDotf/5zKi7VtxAklSOptXP45/tzgLgspGDWf/nKZETSZknp6RPDCEcBExPkmRsCGGP1X1+\nkiSDgcEArVu3dixQGavm0iXc8OwgcguX8PDW+/Lixm3/5/78ghFl+n5l/XqSVBLDN9+DPb4ZwyGf\nvcZN/x7Iocdfy+KcGrFj/cmKPjMn9mtfwUmkslWakYCdgYNDCBOBh0kvA7oRaBBC+K1ctACmliqh\nVM11feN+Np/+LRMbNOWKvU+PHUeSKkYIXLrvuUxaswlb/vg1PV+7L3YiKaOUuAQkSdIrSZIWSZLk\nA0cDryRJchwwCji8+GEnAk+XOqVUTe32zVjOeu9JloYsuhx0EfNr1o4dSZIqzNzcNeh8cA+WZGVz\n6pin2fPr92NHkjJGeewT0BPoGkKYQDpHYEg5vIdU5eXNncl1I64D4IZdjuXD5ptGTiRJFe+jZpsw\naNcTALh2xPWsPefnyImkzFAmJSBJkleTJDmo+PibJEm2T5JkwyRJjkiSZFFZvIdUnWQVFXLjv6+l\n8fxfeWPdbbhtxyNiR5KkaO7c4VDeWHcbGi2YzfUjBpFVVBg7klTtuWOwFMF57zzKTpM+ZkadBnQ9\n6CKKsrJjR5KkaJKQRdeDuvLTGmuy83cfc9boJ2JHkqo9S4BUwXac9DEXvPUQRQQuPKiby4FKEjCj\nbkO6HdgFgK5vPADvvhs5kVS9WQKkCtRw/q/c+O9ryU6KuKXtkbyVv03sSJJUaby6QWvuanMIOUkR\nHHMMzJoVO5JUbVkCpAoSkiKue/Y6msydyegWW3DjLsfGjiRJlc7A3U7kkyYbwMSJcMopkLiVkFQe\nLAFSBTnjvSfZ49uxzKxdnwv+0Z1C5wFI0p8szqnBeR16Qv368NRTMGhQ7EhStWQJkCpAq6nj6f7a\nMAAuat+FH+o3jpxIkiqv79ZqBsPSz0wKCuD11+MGkqohS4BUzhrP+4Vbh/cjJynizu0PZdQGbWJH\nkqTKr0MH6NEDCgvhqKNg2rTYiaRqxRIglaOcwqXc8nR/ms79mfdabM7A3TrFjiRJVcfVV8Puu8MP\nP6RFYMmS2ImkaiMndgCpOit49V52nDyOH+s25NwOvVia7T85SVplOTnw8MPQqhW88QZcfDEMHFjm\nb5NfMKLMX1Oq7BwJkMrJwZ+9xmljnmZxVg5nd+jlfgCSVBJ/+xs8+ihkZ8O118KTT8ZOJFULlgCp\nHGw6/Vv6P38TAFfsfToftNgsciJJqsJ22eX3EYCTToIvv4waR6oOLAFSGau/cC53PNWX2ksX8fiW\ne/PAtgfGjiRJVd+FF8Lhh8OcOXDYYTB3buxEUpVmCZDKUlERN/z7WvJnTWNckw24ZN9zIITYqSSp\n6gsBhgyBTTaBceOgUycoKoqdSqqyLAFSWbriCvb6Zgy/1KrHWR0vZlGN3NiJJKn6qF8fnn4a1lwz\n3Ujs8stjJ5KqLEuAVFaGD4c+fSgi0Png7kxZs0nsRJJU/WyyCTzyCGRlwZVXppOGJa02S4BUFj78\nEI47DoCBu3fijfVaRQ4kSdXYfvvBoEHp8UknwQcfRI0jVUWWAKm0vv8e/vEPmD8fOnXi9h0Oj51I\nkqq/Cy6AU06BBQvS3YV/+CF2IqlKsQRIpTF/Phx8MEydmi5hN3iwE4ElqSKEALfdBjvvDFOmQMeO\nsHBh7FRSlWEJkEqqqChdnWLsWFh//XSSWq4TgSWpwuTmppuHrbMOvPsunHkmJEnsVFKVYAmQSqp3\nb3jiiXSVimefhcaNYyeSpMyz9trpikFrrAHDhv2+qZikv5QTO4BUJd1/P/Ttm25j/+ijsJk7AktS\nRckvGPGnc/vteyF3Du8LPXumIwNHH73S50iZzJEAaXW9+Sacdlp6fPPNsO++cfNIkvjPJjtx9R6n\npDdOPBFefTVqHqmyswRIq+OLL+CQQ2DxYujcGc4+O3YiSVKxu7bvCOefn35GH3IIfPpp7EhSpWUJ\nkFbV1Knpt/4//wzt2/++RrUkqXIIAa6/Hg49FH79FQ44IP3slvQnlgBpVcyaBfvvD5MmwY47pvMA\ncpxSI0mVTnY2PPAA7LQTTJ4MBx4Is2fHTiVVOpYAaWUWLEj3Ahg3Lp0A/Oyz6SoUkqTKqXZteOYZ\n2GQT+PhjOPRQahQuiZ1KqlQsAdJfWboUjj0W3ngDmjeH//wHGjWKnUqStDKNGsHzz0OTJjByJP2f\nv8k9BKRlWAKkFUkSOOccGD4cGjRIC0DLlrFTSZJW1XrrwYgRUKcOh346iktfudsiIBWzBEgrctll\ncNddUKtWegnQFlvETiRJWl3bbQdPPMHirBxOG/M0Xd58MHYiqVKwBEjLc/PNcOWVv28GtvPOsRNJ\nkkpqv/04/+AeLA1ZXPD2Q5w5+vHYiaToLAHSH915Z7oHwG/H//hH3DySpFL7zyY70a19F4oI9Hr1\nPk744NnYkaSoLAHSsu65B846Kz2+6SY49dS4eSRJZWb4Fnty6X7nAHDlS3dw+CcvR04kxeNC5yoT\n+QUjVnjfxH7to2f4K//NN2wYnHZaejxoULrrpCSpWnlwmwOovXghvUcNof/zN7EwpybPbrZb7FhS\nhXMkQAJ46CE4+eR01Yh+/aBr19iJJEnlZMj2Hblul+PIToq4/tlB7D1hdOxIUoWzBEiPPQYnnABF\nRelk4J49YyeSJJWzm3Y6mjt2OIwaRYXcNvwai4AyjiVAGW3fL99JNwMrLITeveHSS2NHkiRVhBDo\nt/tJ3LvdP8gtXModT/Vl/y/eip1KqjCWAGWs/b94i1ue7p/uClxQAH36xI4kSapIIdBn7zO4c/tD\nqVFUyC1P9+fgz16NnUqqEJYAZaTDP3mZW5/uT82ipdCtG/TtCyHEjiVJqmghcM0eJ3PjTkeTkxRx\nw78HccTHL8ZOJZU7S4AyzkljnuHa524gOyniul2OgwEDLACSlMlC4Ppdj2fAbp3IImHg8zdx/IfP\nxU4llStLgDJHknD+Ww9x+cjBAPTZ+3Ru2vkYC4AkCYDb2h7JlXulS0Vf9eJtnPr+8MiJpPJT4hIQ\nQmgZQhgVQvgshPBpCOGC4vMNQwgvhRC+Kv59rbKLK5VQknDJqCFc9Oa/KAxZdD+gM/e27hA7lSSp\nkhnS5hAubXc2AL1fuZvz33ooXT5aqmZKMxKwFLgoSZLNgR2Bc0MImwMFwMgkSTYCRhbflqLJKiqk\n3ws3c/r7w1mclcP5B/fgsa33jR1LklRJPdCqPd0P6EwRgYve/BdXvXgb2UWFsWNJZarEJSBJkmlJ\nknxQfDwHGA80BzoAQ4sfNhQ4pLQhpZLKXbqYm54ZyNEfv8iCnFxOP6w3z226S+xYkqRK7rGt9+Xs\nQ3qxMKcmx3/0PHc8dTW1liyMHUsqM2UyJyCEkA9sC4wGmiRJMq34rh+AJmXxHtLqajj/V/718CUc\n9MWbzKlZm05H9uG19beLHUuSVEX8Z5OdOO6oq5hVqy7tJrzHQw9dQsP5v8aOJZWJnNK+QAihLvAE\ncGGSJLPDMpMskyRJQgjLvZAuhHAGcAbAOuusU9oYqgD5BSNiR1hl6/88hXse70P+rGl8X6/x/7d3\n5/FRVXcfxz8nC2EPEAKEAAkYZFEUSsQqUkGgIiq7j1SkihVr0fZ50Fbp4660xSpWfdra8iqgqEUU\ntMjigguLCwooht2EQNgJayBACITz/HEukEBASGZyJzPf9+t1X3Pn3GXO/eXkzD1z77mHOwY9xuoG\nzf3OloiIBFmgv6uWNGnLwFuf4ZU3H6PD1jVMfe133H6TxpWRyq9cVwKMMbG4BsDr1tq3veTtxpgk\nb3kSkFvattbacdbadGttemJiYnmyIVJCp43Lefu135K6dyvLG15Av6Fj1QAQEZEyW5vQlP5Dn2VF\ngxa02LOFaa/9DhYt8jtbIuVSnqcDGWA8sMpa+1yxRe8Ct3nztwHTy549kfPTd8WnvDrlYeoU5DMn\nrRP/dcsYcmsl+J0tERGp5HbUrMfNt4xhfmoHEg/uha5d4T96hKhUXuW5EtAZGApcY4xZ6k29gTFA\nT2NMJtDDey8SXNbym88n88LMscQVHWVixxv5Zf+HOFilmt85ExGRMJEfV507Bj3GtIuvgYMHoX9/\neOIJOHbM76yJnLcy9wmw1n4GnGmUpe5l3a/IeTtwAIYP577PJnMMw5Pdh/Nyeh+/cyUiImHoaHQM\n9/ceycCh18KoUfD447B0KUyaBLVq+Z09kXOmEYOlcsvMhB//GCZP5kBsVe4a8LAaACIiElzGwAMP\nwKxZEB/vbgu64gpYu9bvnImcMzUCpPKaMQMuuwyWL4dWrejz87/wUcvL/c6ViIhEiuuug6+/htat\nYcUK9500Z47fuRI5J2oESOVTVASPPAJ9+kBenrsn8+uvWVu/qd85ExGRSHPhhfDVV3DjjbBnD/Tq\nBWPHgi31CekiIUONAKlcdu+GG26A0aMhKgrGjIFp06B2bb9zJiIikap2bXdL0MMPu07Cv/0t9O0L\nu3b5nTORM1IjQCqPhQshPR3efx8SEuCDD+DBB929mSIiIn6KioKnnoK334Y6ddwtq+3bw/z5fudM\npFRqBEjoKypyFetVV8G6ddCxIyxZAj16+J0zERGRkvr3d08LuuIK2LQJunWDJ59032UiIUSNAAlt\n69e7AVkefdRVoPffD59/DikpfudMRESkdCkpMG8e/P73rm/AY4+5H662bPE7ZyInlHmcAAlfqaNm\nVcj+1o+5/uwbTp4Md98N+/ZBUpJ7BrN+/RcRiQiB/i6qcLGx8Mc/uisBt94Kc+fCpZdy509+VeqT\n7H7wO7GSKvM5gASdrgRI6Nm3D4YOhVtucfN9+0JGhhoAIiJS+fTsCd9951537uRfbz/F2JljqV2Q\n73fOJMKpESCh5b33oF07eO01qF4dxo2Dd96B+vX9zpmIiEjZNGrkHmrx3HMUxFRh4IpPmTN+BN2z\nvvI7ZxLB1AiQ0LBjh7tc2rs3bNjgOv9+8w0MH66n/4iISOUXFQUjR3LdsP9jUXJbGubvZvy0p3hu\n5ljiD+33O3cSgdQIEH9ZC6+/Dm3butdq1eCZZ9zjQFu18jt3IiIiAbWuXjI33/InnrxmOIdi4hjg\nXRVg+nS/syYRRo0A8U9ODlx/vbsCsHMnXHMNLFvmBlmJUZ91EREJT8eioplwWV+uG/YiXzdpS4MD\ne6BfPxgwwH03ilQANQKkwsUWHWH4V2/DRRe5PgB16sD48fDRR3DBBX5nT0REpEKsr5fMzbeM4Ynu\nw6FGDdcHrk0bGD0aCgr8zp6EOTUCpOJYS/esr/hw/AgemjsBDhyAQYNg1Sq44w7d+y8iIhHHmigm\npj1DLz8AABEsSURBVPeFNWtg8GA4dAgeeQQuvhhmz/Y7exLG1AiQCpG2cwOT3nyU8dOeovmerWTV\na+KuArz1lntqgoiISCRLTnbj43zyiesnt3atu2W2Tx/IzvY7dxKG1AiQoIo/tJ/HPvon70+4l5+s\n/5a8uBo80X04ve74K/Tq5Xf2REREQku3brB0KTz7LNSsCTNmuFuE7rvP9Z8TCRA1AiQoqhUWcPfC\nqcwddxfDlszAAK926E3Xu8YxMb0vR6PV8VdERKRUsbFw//3uFqEhQ6CwEP7yF9dvbvRoyNdAY1J+\nagRIQMUdLWTY4unM/+edjJr3MnUL9vNFs0u4/vYXeOSnI9hTPd7vLIqIiFQOjRu7wTO/+QauvRb2\n7XP9BdLS4O9/hyNH/M6hVGL6OTYMpI6aVWr6+jHXV1geYouOcNOyj/j152+QlL8LgKVJLRnbZSgL\nUjuU2un3TPmGis17WeJ3tryLiEj4q8jvsNQpW6D9r7mi7tU8OO9l2m/9Hu65h/UPjSb1xaddh+LY\nWN/yVxahnr9IoEaAlEuVo0fou/JTfv3FFJrlbQdgVWIqY7sM5aO0Tnrij4iISIB8mXIJ/YaO5drv\nv+SB+ZO4YPcm+PnP4dFH4YEHYNgwqFrV72xKJaFGgJRJ7YJ8bln6PsOWvEvD/N0ArK3XhOeuGsLs\n1p2xRneaiYiIBJwxfNDqSj5qeTn9V3zKs1mz4fvvYcQIePJJ14H47rv9zqVUAjpTk/PSaN9O/veT\n8Xz+0jBGzXuZhvm7WV0/hZHX38dPf/E3ZrXpogaAiIhIkBVFRTO1XQ9YudI9brtDB9i2zV0RSElh\n5ILXSczf43c2JYTpSoCck4u3ZTFsybv0WTmP2GNFAHzR7BLGdRrA3BYddduPiIiIH6Kj3cCbAwfC\nBx/AH/4An33Gf38xmV8tfItZra9i0o9u4NvGrfRdLSWoESBnVK2wgBtXzWfI0ve4dFsmAEUmipmt\nu/DPTgNYltTS5xyKiIgI4E7we/Vy04IFfPCLB+iR9TX9V86l/8q5ZDRKY9KPbmBG6y5+51RChBoB\ncpqWO3IYsvQ9Biz/hNqFBwHIi6vBtIu7MzG9DxvraIRfERGRkNWlC78c8DDJebnc+u1sbs74kEu2\nZfHs7Of5308nQPRCuP12NzKxRCw1AsTZvRumToVJk5jz+ecnkr9p3IrX2/dmZuurOBwb52MGRURE\n5Hxsjm/A011v5/nOP+PG1Qu4bckM2m1fC88846bLLnONgcGDoV49v7MrFUyNgEhWUAAzZ8Lrr8Os\nWScGHcmvUo13LurGv9v3YlWDFj5nUkRERMrjcGwcU9v1YOrF3emwZQ3vVFsDU6bAokVuGjkS+vSB\n225zg5KdMuaAhCc1AiLNkSMwd6775586FfLyXHpUFPTsCbfeyuXfVuNAXHVfsykiIiIBZgzfJreG\nMffDCy/Af/4Dr7wCH37ozgmmTnVXBPr2dZ2Ne/SAKlX8zrUEiRoBkSA/3z0x4J133C/+e/eeXNax\nIwwZ4i4FJiUBcGClRsMVEREJa9Wqwc9+5qbNm+G112DSJPfI0YkT3RQf764QDBzorhBoILKwokZA\nmGqwfxdMmOBO/OfMgcOHTy686CLo39+d/Ldu7V8mRURExH/JyfDgg25auRKmTXNXBTIy4NVX3VSj\nhrsycN11bmrWzO9cSzmpERAmqh4poNPGFXRZ/y1d1n1L6505JxcaA1deCf36uamlHu0pIiIipWjb\n1k2PPOJGIp42zU1LlsD06W4C94Pi8QbBVVfptqFKyFhr/c4D6enpdvHixX5nw3epo858G876MdeX\nTCgsdP+QCxaw4O+T6bRpBXFFR04sPhBblRo/7e4u4/XpA43O/bGeZ8uHiIiIBN5p3/PFnNf5wTls\nU5Z8XDHiZa7OXkK37MV0zvmOmoWHTi6sXh06d4arr4auXd1Th7xGQUWdU4R6/CqSMWaJtTb9h9bT\nlYDKIj8fvvwSFixw01dfwSH3D9gFOIYho1EaC1I7sKB5B5YktyHzmX7+5llERETCwtbaibzRvhdv\ntO9FbNER0jet4ursxXTNXuLuPpgzx03g+htceSV07coVOVFkNGqpB46EIDUCQlBM0VFa7cyh3dZM\nLtmWyaVbM+HZHCgqKrlimzbQpQu/2RrPZ6nt2V093p8Mi4iISMQ4Eh3LlymX8GXKJYzpdgfrR6bD\n/Pnu6YPz5sGKFfDxx/Dxx0zG/VD5ff1mLG3cim8bt2Jp41ZkJjTlWFS034cS0dQI8NuBA7B8OSxb\nxuNzpnPp1kza5maXuLUHgOhod3mtSxc3de4MiYkAvKvbd0RERMQvDRvCTTe5CSA31zUK5s3ju6kf\n0DY3m9Y7c2i9M4fBGR8C7rbl1YmprGzYgpUNWrCqQXNWJ6ZQEKsnEFUUNQIqyqFDkJkJa9a4k/6M\nDFi2DLKzweuXcXux1dfVTSKj0YVkJLUko1Eab4271/XMFxEREQllDRq4cQYGDaJvjVnEHTnMRduz\n6bB1De23rKH91u9pmredjltW03HL6hObFZko1tVtzOrEVNYmNGFtQlOyEpqytl4yh2PjfDyg8KRG\nQCAVFEBODqxbB1lZ7oR/zRrXu37DhhMn+yXExrrHdLZrx9ObY8lo1JJljdLYV7VmyfXUABAREZFK\n6HBsHN80acM3TdqcSKt7MI82uetom5vtva4jbddG0nZvIm33phLbH8OwKb4BWQlNyambRE6dJHLq\nJrGhTiM2xjeiMEYjHJeFGgHnylo3uu6mTSenjRvdCf/xafPmM28fEwMXXAAXXugeq9WunZtatTrR\ng/4l3dYjIiIiEWBP9Xi+SG3PF6ntT6RVOXqEljtzaLlro2sQ7NpI2s6NpOzdSrO87TTL237afo5h\n2FqrPixsCykp0LSpG8OgWbOT81IqNQIAduyAbdtKTtu3n5zfvNmd9Ofnn30/0dGusDVvDi1auBP8\n41Pz5u5XfxERERE5TWFMLCsapbGiUVqJ9Jiio6Ts3Uraro0027ON1L1baLZnGyl7t9J43w6S9+9w\nHZLPYFmVauTWTCC3Zl1ya9Qr8cqcKu72pcREqF8/osY7UCMA4PLL3S/5P6RGDdeqbNLk5JSaevKk\nv0kT94u/iIiIiATE0egY1iY0ZW1C09OWxRQdJXlfLvMGpbo7NDZscFOx+VoFh6i1exMXnHKbEQAz\nx5Z8Hx8PiYlMOxjD3mq12Fu1JnlVa5FXteaJ9/uq1mR/XHX2x9U48ZpfpRrWRAUnAEEStDNWY0wv\n4AUgGviXtXZMsD6r3Fq3dgNdNGzoBtU6PjVs6KbkZHfyX7u2G31XRERERHx3NDqGnLqNoWfP0lew\nlkv/ZwoN8nfT4MAe73U3DfL30DB/Nzckxbg7QnJzYedOd+t3Xh4dy5CX/VWqQd5t8NJL5TqmihKU\nRoAxJhr4G9AT2AQsMsa8a61dGYzPK7fZs/3OgYiIiIgEmjHkVatFXrVaZCamnLb4huIj/B47Bnv2\nwI4d3PTUdOILDlCnYD/xBfnEH9pPnYJ86hTsp3bBAWodPkCtwwepWXjQzRceolbhodPHdAphwboS\n0AnIstZmAxhj3gD6AqHZCBARERGRyBYVBQkJkJDAoqZrz2/TY0XULDxExuPXBilzgWdsaY+tLO9O\njRkE9LLW3um9Hwpcbq29t9g6dwF3eW9bAWsCnpHzUx/Y6XMewoViGRiKY2AojoGjWAaG4hg4imVg\nKI6BEwqxTLHWJv7QSr71YrXWjgPG+fX5pzLGLLbWpvudj3CgWAaG4hgYimPgKJaBoTgGjmIZGIpj\n4FSmWAarG/NmoHgX7iZemoiIiIiI+CxYjYBFQEtjTHNjTBVgMPBukD5LRERERETOQ1BuB7LWHjXG\n3At8gHtE6ARr7YpgfFYAhcytSWFAsQwMxTEwFMfAUSwDQ3EMHMUyMBTHwKk0sQxKx2AREREREQld\nlWtoMxERERERKTc1AkREREREIkzYNAKMMb2MMWuMMVnGmFFnWW+gMcYaY9K991WMMRONMcuMMd8Z\nY7oWW7ejl55ljHnRGGO89HrGmDnGmEzvtW7QD7ACBTqWxpjqxphZxpjVxpgVxpgxxfZxuzFmhzFm\nqTfdGfQDrCBBKpNzvX0ej1cDLz3OGDPF+6yvjDGpQT68ChWEMlmrWAyXGmN2GmOe95ZFbJk827Eb\nY27z6rxMY8xtxdJVTwYglqonA1omI66eDEJ5jMg6Esody/eNMXuNMTNP2aa5V+ayvDJYxUv3t0xa\nayv9hOt8vBZoAVQBvgPalrJeLWA+sBBI99LuASZ68w2AJUCU9/5r4MeAAd4DrvPS/wyM8uZHAU/7\nHYNQjiVQHejmpVcBFhSL5e3AX/0+7soQR+/93OPrnbKfEcA/vPnBwBS/YxDqsTxl2yXATyK9TJ7p\n2IF6QLb3Wtebr+stUz0ZgFiqngxomYyoejJYcTxlvbCvI8sbS29Zd+BGYOYp6W8Cg735fwC/CoUy\nGS5XAjoBWdbabGttIfAG0LeU9Z4CngYKiqW1BT4BsNbmAnuBdGNMElDbWrvQur/OJKCft01f4BVv\n/pVi6eEg4LG01h601n7qpRcC3+DGjghnAY/jD3xe8TI5Feh+/BfZMBDUWBpjLsQ1EBYEPush5Vzj\nWJprgTnW2t3W2j3AHKCX6snAxVL1ZGDi+APbhGs9GdQ4RlAdCeWLJdbaj4H9xdO8MnYNrsxByfrQ\n1zIZLo2AZGBjsfebvLQTjDE/Appaa2edsu13QB9jTIwxpjnQETfQWbK3n9L22dBau9Wb3wY0DMhR\nhIZgxLL4tnVwreSPiyUPNMZkGGOmGmNKrF+JBTOOE71LkI8UqyxOfJ619iiQByQE7Gj8FdQyyclf\nX4o/Ki0iy6SntGM/07aqJ51AxPIE1ZOnKUscI6meDGp5JHLqSChfLM8kAdjrlblT9+lrmQyXRsBZ\nGWOigOeA+0tZPAH3B1kMPA98ARSd6769f4qIec5qeWJpjIkBJgMvWmuzveQZQKq19hLcLxCvEAHK\nEcch1tp2QBdvGhr83Ia2APx/D8aVy+Miskx6gnLskVZPesoUS9WTpynLsauePF15ypDqyJLC5vjD\npRGwmZK/7jXx0o6rBVwMzDXGrMfdv/quMSbdWnvUWjvSWtveWtsXqAN8723f5Az73O5dBsd7zQ3C\nMfklGLE8bhyQaa19/niCtXaXtfaw9/ZfuF9qw0FQ4mit3ey97gf+jbt0WeLzvJOIeGBXkI6togWt\nTBpjLgVirLVLjqdFcJk827GfaVvVk04gYnmc6sliyhLHCKwng1YeI6yOhPLF8kx2AXW8MnfqPv0t\nk+fbiSAUJ9zIx9lAc0525LjoLOvP5WTHwepADW++JzC/2Hqndnjr7aU/Q8kOb3/2OwaVIJajgWmc\n0ikTSCo23x9Y6HcMQjWO3j7re/OxuPsH7/be30PJzkVv+h2DUI5lsXXHAE+oTJ792HGdBtfhOg7W\n9ebrectUTwYulqonyxnHSKwng1UeveURU0eWN5bF0rpyesfgtyjZMXhEKJRJ3wMewD9cb9wvfGuB\nh7y0J4E+paw7l5MnCanAGmAV8BGQUmy9dGC5t8+/cnKE5QTcvZqZ3jb1gnVc4RBLXKvXeulLvelO\nb9mfgBXeP9qnQGu/jz+E41gD94SGDC9mLwDR3rKqXiWThTspa+H38YdyLIutm31qmYvkMnm2Ywfu\n8MpXFjCsWLrqyQDEUvVkwOIYkfVkMP63vWURVUcGIJYLgB3AIdytqNd66S28MpfllcG4UCiTxytr\nERERERGJEOHSJ0BERERERM6RGgEiIiIiIhFGjQARERERkQijRoCIiIiISIRRI0BEREREJMKoESAi\nIiIiEmHUCBARERERiTD/DwR1CWvVh9FXAAAAAElFTkSuQmCC\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 100000\n", "N_trials = 1000\n", "result = np.random.randn(N, N_trials) + .5\n", "\n", "mean = result.mean(axis=0).mean()\n", "std = result.mean(axis=0).std()\n", "bins = np.linspace(0.49, .51, 100)\n", "\n", "fig, ax = plt.subplots(figsize=(13, 8))\n", "ax.hist(result.mean(axis=0), bins=bins, normed=True)\n", "ax.plot(bins, 1 / np.sqrt(2*np.pi) / std * np.exp(-.5*(bins-mean)**2/std**2), 'r', lw=2 );" ] } ], "metadata": { "kernel_info": { "name": "python3" }, "kernelspec": { "display_name": "Python 3", "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": "3.6.0" } }, "nbformat": 4, "nbformat_minor": 1 }