{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Coin-flip experiment and the Central Limit Theorem\n", "\n", "Question: I flip a fair coin 100 times. What is the probability distribution of the number of heads \n", "that I get." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from scipy import stats\n", "\n", "import matplotlib as mpl # As of July 2017 Bucknell computers use v. 2.x \n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "# As of Aug. 2017 M.L. reverting to 1.x defaults.\n", "# In 2.x text.ustex requires dvipng, texlive-latex-extra, and texlive-fonts-recommended, \n", "# which don't seem to be universal\n", "# See https://stackoverflow.com/questions/38906356/error-running-matplotlib-in-latex-type1cm?\n", "mpl.style.use('classic')\n", " \n", "# M.L. modifications of matplotlib defaults using syntax of v.2.0 \n", "# More info at http://matplotlib.org/2.0.0/users/deflt_style_changes.html\n", "# Changes can also be put in matplotlibrc file, or effected using mpl.rcParams[]\n", "plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures\n", "plt.rc('axes', labelsize=16, titlesize=14)\n", "plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Simulating spin flips\n", "\n", "I am going to generate 100 random 0's and 1's, and let 1 represent a \"heads\" and 0 represent a \"tails.\"" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[0 1 1 0 1 0 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0\n", " 0 1 1 1 0 0 1 0 0 1 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0\n", " 0 1 1 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0]\n" ] } ], "source": [ "n_flips = 100 # Number of spin flips\n", "data = stats.randint.rvs(0, 2, size=n_flips)\n", "print(data)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "I can count the \"heads\" by summing the array `data`. " ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "54\n" ] } ], "source": [ "n_heads = np.sum(data)\n", "print(n_heads)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Simulating many trials of 200 spin flips\n", "I store the number of heads in each trial in the array `results`." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "n_expts = 200 \n", "results = np.zeros(n_expts, dtype=int) # Create array in which to store results of experiments\n", "\n", "for i in range(n_expts):\n", " results[i] = np.sum(stats.randint.rvs(0, 2, size=n_flips))" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n", " 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n", " 0, 0, 0, 0, 1, 1, 2, 3, 6, 6, 6, 8, 13, 14, 22, 19, 23,\n", " 19, 15, 9, 9, 2, 6, 7, 5, 1, 2, 1, 0, 0, 0, 0, 0, 0,\n", " 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n", " 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],\n", " dtype=int32)" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "test = np.bincount(results,minlength=n_flips)\n", "test" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "np.bincount?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Histogram of results" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "/lig" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
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')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "39\n" ] }, { "data": { "text/plain": [ "(array([3.000e+00, 9.000e+00, 2.300e+01, 5.600e+01, 7.700e+01, 1.540e+02,\n", " 2.760e+02, 4.280e+02, 7.100e+02, 1.131e+03, 1.634e+03, 2.185e+03,\n", " 2.994e+03, 3.922e+03, 4.869e+03, 5.729e+03, 6.699e+03, 7.239e+03,\n", " 7.768e+03, 7.911e+03, 7.795e+03, 7.414e+03, 6.720e+03, 5.934e+03,\n", " 4.949e+03, 3.819e+03, 2.949e+03, 2.151e+03, 1.551e+03, 1.098e+03,\n", " 7.400e+02, 4.440e+02, 2.590e+02, 1.700e+02, 9.600e+01, 5.000e+01,\n", " 1.800e+01, 7.000e+00, 1.000e+01]),\n", " array([30.5, 31.5, 32.5, 33.5, 34.5, 35.5, 36.5, 37.5, 38.5, 39.5, 40.5,\n", " 41.5, 42.5, 43.5, 44.5, 45.5, 46.5, 47.5, 48.5, 49.5, 50.5, 51.5,\n", " 52.5, 53.5, 54.5, 55.5, 56.5, 57.5, 58.5, 59.5, 60.5, 61.5, 62.5,\n", " 63.5, 64.5, 65.5, 66.5, 67.5, 68.5, 69.5]),\n", " )" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "low = 30.5\n", "high = 69.5\n", "nbins = int(high - low)\n", "plt.figure()\n", "plt.xlabel(\"# of heads\")\n", "plt.ylabel(\"occurrences\")\n", "out = plt.hist(results, nbins, [low,high])\n", "plt.xlim(low,high);\n", "print(nbins)\n", "out" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Checking Central Limit Theorem\n", "Following Hughes and Hase statement of the Central Limit Theorem at the \n", "top of p.33, we should look at the distribution of the sample mean:\n", "\n", "$$\n", "\\langle x\\rangle = \\frac{1}{N}(x_1 + x_2 + \\cdots + x_N).\n", "$$\n", "\n", "It's the distribution of the sample mean that approaches the normal distribution.\n", "\n", "The individual values $x_i$ are sampled from a discrete distribution with two values:\n", "$x_i = 0$ with a probability of 0.5, and $x_i = 1$ with a probability 0.5. The \n", "mean of this parent distribution for the $x_i$'s is $\\langle x\\rangle = 0.5$, and the \n", "standard deviation of this parent distribution is $\\sigma = 0.5$.\n", "\n", "The histogram below is the same as the histogram immediately above, except that the \n", "results for the number of heads in each trial have been divided by $N$, the number of \n", "coin-flips in a trial (in this example 100).\n", "\n", "The Central Limit Theorem states that total number of heads divided by the number of \n", "flips (100) should tend toward a normal distribution with a mean of 0.5 and a standard deviation of $\\sigma_\\text{parent}/\\sqrt{100}$. Let's check." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([48., 52., 53., ..., 50., 48., 51.])" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "results" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "nbins = 41\n", "low = 0.3\n", "high = 0.7\n", "\n", "plt.figure()\n", "plt.xlabel(\"(# of heads)/100\")\n", "plt.ylabel(\"occurrences\")\n", "h_out = plt.hist(results/n_flips, nbins, [low,high], alpha = 0.5)\n", "plt.xlim(low,high);\n", "\n", "x = np.zeros(len(h_out[1])-1) # Create array for mid-points of histogram bins\n", "y = np.zeros(len(h_out[1])-1) # Create array for expected occurences from normal dist.\n", "for i in range(len(x)):\n", " x[i] = (h_out[1][i+1] + h_out[1][i])/2\n", " y[i] = n_expts*(stats.norm.cdf(h_out[1][i+1],0.5,0.5/np.sqrt(n_flips))\\\n", " - stats.norm.cdf(h_out[1][i],0.5,0.5/np.sqrt(n_flips)))\n", " \n", "plt.scatter(x, y, c = 'red');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Looks pretty Gaussian!!\n", "\n", "In a future class we will talk about how to make quantitative assessments of confidence \n", "about whether a sample of random variables comes from a given distribution." ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "aa = np.linspace(30,70,21)" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0.2970297 , 0.31683168, 0.33663366, 0.35643564, 0.37623762,\n", " 0.3960396 , 0.41584158, 0.43564356, 0.45544554, 0.47524752,\n", " 0.4950495 , 0.51485149, 0.53465347, 0.55445545, 0.57425743,\n", " 0.59405941, 0.61386139, 0.63366337, 0.65346535, 0.67326733,\n", " 0.69306931])" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "aa/101" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([0.3 , 0.31904762, 0.33809524, 0.35714286, 0.37619048,\n", " 0.3952381 , 0.41428571, 0.43333333, 0.45238095, 0.47142857,\n", " 0.49047619, 0.50952381, 0.52857143, 0.54761905, 0.56666667,\n", " 0.58571429, 0.6047619 , 0.62380952, 0.64285714, 0.66190476,\n", " 0.68095238, 0.7 ])" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h_out[1]" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "ename": "ValueError", "evalue": "operands could not be broadcast together with shapes (21,) (22,) ", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0maa\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0;36m101\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mh_out\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;31mValueError\u001b[0m: operands could not be broadcast together with shapes (21,) (22,) " ] } ], "source": [ "aa/101 - h_out[1]" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "46.97674418604651" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(70-30)*n_flips/2/nbins" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.4697674418604651" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(high-low)*n_flips/2/nbins" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(array([4.000e+00, 1.400e+01, 1.700e+01, 2.600e+01, 0.000e+00, 6.800e+01,\n", " 1.110e+02, 2.060e+02, 3.060e+02, 5.660e+02, 9.590e+02, 1.334e+03,\n", " 1.840e+03, 2.536e+03, 3.557e+03, 4.419e+03, 5.374e+03, 6.342e+03,\n", " 7.128e+03, 7.447e+03, 7.838e+03, 0.000e+00, 7.841e+03, 7.635e+03,\n", " 7.018e+03, 6.362e+03, 5.227e+03, 4.227e+03, 3.380e+03, 2.609e+03,\n", " 1.955e+03, 1.340e+03, 9.010e+02, 5.840e+02, 3.380e+02, 2.280e+02,\n", " 1.260e+02, 6.800e+01, 0.000e+00, 3.800e+01, 1.500e+01, 1.000e+01,\n", " 1.000e+00]),\n", " array([0.3 , 0.30930233, 0.31860465, 0.32790698, 0.3372093 ,\n", " 0.34651163, 0.35581395, 0.36511628, 0.3744186 , 0.38372093,\n", " 0.39302326, 0.40232558, 0.41162791, 0.42093023, 0.43023256,\n", " 0.43953488, 0.44883721, 0.45813953, 0.46744186, 0.47674419,\n", " 0.48604651, 0.49534884, 0.50465116, 0.51395349, 0.52325581,\n", " 0.53255814, 0.54186047, 0.55116279, 0.56046512, 0.56976744,\n", " 0.57906977, 0.58837209, 0.59767442, 0.60697674, 0.61627907,\n", " 0.6255814 , 0.63488372, 0.64418605, 0.65348837, 0.6627907 ,\n", " 0.67209302, 0.68139535, 0.69069767, 0.7 ]),\n", " )" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "h_out" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.0046511599999999875" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "0.5-0.49534884" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "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": "3.5.5" } }, "nbformat": 4, "nbformat_minor": 2 }