diff --git a/Charting Coin Flips.ipynb b/Charting Coin Flips.ipynb new file mode 100644 index 0000000..2315014 --- /dev/null +++ b/Charting Coin Flips.ipynb @@ -0,0 +1,228 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:2b10286201a4c67d650b3529f96453faab729b2bee426b0661a05e7abd50dea9" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "Charting Coin Flips: Draft Notes" + ] + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "I still have much proving to do." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "However, I understand that this assignment gives us a look into Bernoulli's Trials:\n", + "\n", + "http://en.wikipedia.org/wiki/Bernoulli_trial\n", + " \n", + "As a coin is flipped repeatedly, the probability of success (landing heads) is 1/2.\n", + "\n", + "It's arbitrary to look at this as mere flipping of a coin because one could also plot the data with random boolean values. Or, random binary data.\n", + "\n", + "Finally, I digress even further into experimenting with NumPy. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let y be a Bernoulli trial:\n", + "\n", + "y\u223cBernoulli(\u03b8)=Binomial(1,\u03b8)\n", + "\n", + "The probability density, or marginal likelihood, function, is:\n", + "\n", + "p(y|\u03b8)=\u03b8y(1\u2212\u03b8)1\u2212y={\u03b81\u2212\u03b8y=1y=0}" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import math\n", + "import random\n", + "import matplotlib.pyplot as plt\n", + "import statistics as stat\n", + "import seaborn as sns\n", + "import numpy as np" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 28 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "np.random.seed(123)\n", + "\n", + "nobs = 100\n", + "theta = 0.3\n", + "Y = np.random.binomial(1, theta, nobs)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 40 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def coin_flip(size):\n", + " xmin, xmax, dx = 1, size, 1\n", + " x = np.arange(xmin, xmax, dx)\n", + " flips = np.random.randint(0, 2, size=size)\n", + " return x, [flips[:i].mean() for i in x]\n", + "\n", + "x, y = coin_flip(100000)\n", + "plt.plot(x, y)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 29, + "text": [ + "[]" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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uv+/5vD4vPBwOc8dPXuCUjQ1ccfZ6bv3X5+Zd0zM4TktDZXzNecxf3Pkof//R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\nnxXpDQ8MT7Dr1cimGZWeMgI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+ "text": [ + "" + ] + } + ], + "prompt_number": 30 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def coin_flip(size):\n", + " xmin, xmax, dx = 1, size, 1\n", + " x = np.arange(xmin, xmax, dx)\n", + " flips = np.random.randint(0, 2, size=size)\n", + " return x, [flips[:i].mean() for i in x]\n", + "\n", + "x, y = coin_flip(100000)\n", + "plt.plot(x, y)" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 31, + "text": [ + "[]" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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+ "text": [ + "" + ] + } + ], + "prompt_number": 31 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "def coin_flip(size):\n", + " flips = np.random.randint(0, 2, size=size)\n", + " return flips.mean()\n", + "coin_flip = np.frompyfunc(coin_flip, 1, 1)\n", + "\n", + "xmin, xmax, dx = 1, 100000, 1\n", + "x = np.arange(xmin, xmax, dx)\n", + "y = coin_flip(x)\n", + "plt.plot(x, y)\n", + "\n", + "\n" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "pyout", + "prompt_number": 36, + "text": [ + "[]" + ] + }, + { + "metadata": {}, + "output_type": "display_data", + "png": 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