classical_random_walk.ipynb

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    "\"\"\"\n",
    "Discrete random walk on a circle\n",
    "\n",
    "Parameters\n",
    "----------\n",
    "\n",
    "p   :  Left-right bias parameter\n",
    "x_0 :  Initial position on the circle\n",
    "n   :  Range of points on the circle are 0, ..., n - 1\n",
    "t   :  Number of steps\n",
    "m   :  Number of times the random walk is performed to generate a probability distribution of final location\n",
    "\"\"\"\n",
    "\n",
    "p   = 0.5\n",
    "x_0 = 1\n",
    "n   = 7\n",
    "t   = 200\n",
    "m   = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "from matplotlib import pyplot as plt\n",
    "from matplotlib import animation\n",
    "\n",
    "%matplotlib inline\n",
    "%matplotlib notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Random walk on a circle with n points\n",
    "\"\"\"\n",
    "class RandomWalk:\n",
    "    def __init__(self, p_step_left, x_0, n, t, samp_num):\n",
    "        \"\"\"Initialize parameters\n",
    "        \n",
    "        :param step_left_prob: probability of stepping left\n",
    "        :param x_0: initial position the circle with values in 0, ..., n - 1\n",
    "        :param n: number of points on the circle and the positions take \n",
    "                  values in 0, ..., n - 1\n",
    "        :param t: timelike parameter that denotes number of steps\n",
    "        :param sample_num: number of samples where random walk is performed\n",
    "        \"\"\"\n",
    "        \n",
    "        self.p_step_left = p_step_left\n",
    "        self.x_0 = x_0\n",
    "        self.n = n\n",
    "        self.t = t\n",
    "        self.samp_num = samp_num\n",
    "        \n",
    "        # Matrix of random walk samples...rows correspond to different \n",
    "        # samples while columns correspond to time steps\n",
    "        self.random_walks = None\n",
    "        \n",
    "        # Position distribution matrix as a function of time steps...rows \n",
    "        # correspond to time steps while columns correspond to position \n",
    "        # on the circle\n",
    "        self.position_dist = None\n",
    "        \n",
    "    def _next_position(self, x):\n",
    "        \"\"\"direction 0 - right, direction 1 - left \"\"\"\n",
    "        \n",
    "        direction = np.random.binomial(1, self.p_step_left)\n",
    "        if direction == 0:\n",
    "            return (x + 1) % self.n\n",
    "        else:\n",
    "            return (x - 1) % self.n\n",
    "        \n",
    "    def _generate_random_walk(self):\n",
    "        walk = [-1] * (self.t + 1)\n",
    "        x = self.x_0\n",
    "        for i in range(self.t + 1):\n",
    "            walk[i] = x\n",
    "            x = self._next_position(x)\n",
    "        return walk\n",
    "    \n",
    "    def _generate_random_walk_samples(self):\n",
    "        self.random_walks = [[]] * self.samp_num\n",
    "        for i in range(self.samp_num):\n",
    "            self.random_walks[i] = self._generate_random_walk()\n",
    "                    \n",
    "    def generate_position_distribution(self):\n",
    "        if self.random_walks is None:\n",
    "            self._generate_random_walk_samples()\n",
    "            \n",
    "        self.position_dist = \\\n",
    "            [[0 for i in range(self.n)] for j in range(self.t + 1)]\n",
    "        \n",
    "        for t_i in range(self.t + 1):\n",
    "            for samp_j in range(self.samp_num):\n",
    "                position = self.random_walks[samp_j][t_i]\n",
    "                self.position_dist[t_i][position] += 1\n",
    "                \n",
    "        for t_i in range(self.t + 1):\n",
    "            for pos_j in range(self.n):\n",
    "                self.position_dist[t_i][pos_j] /= (1.0 * self.samp_num)\n",
    "                \n",
    "                \n",
    "    def animate_position_distribution(self):\n",
    "        fig = plt.figure()\n",
    "        ax = plt.axes(xlim=(0, self.n), ylim=(0, 1.05))\n",
    "        line, = ax.plot([], [], lw=2)\n",
    "        \n",
    "        def init():\n",
    "            line.set_data([], [])\n",
    "            return line,\n",
    "        \n",
    "        def animate(i):\n",
    "            x = list(range(self.n))\n",
    "            y = self.position_dist[i]\n",
    "            line.set_data(x, y)\n",
    "            return line,\n",
    "        \n",
    "        anim = animation.FuncAnimation(fig, animate, init_func=init,\n",
    "                                       frames=self.t + 1,\n",
    "                                       interval=1000, blit=True)\n",
    "        plt.show()"
   ]
  },
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   "execution_count": 6,
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       "mpl.figure.prototype.handle_figure_label = function (fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function (fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch (cursor) {\n",
       "        case 0:\n",
       "            cursor = 'pointer';\n",
       "            break;\n",
       "        case 1:\n",
       "            cursor = 'default';\n",
       "            break;\n",
       "        case 2:\n",
       "            cursor = 'crosshair';\n",
       "            break;\n",
       "        case 3:\n",
       "            cursor = 'move';\n",
       "            break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_message = function (fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function (fig, _msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n",
       "    for (var key in msg) {\n",
       "        if (!(key in fig.buttons)) {\n",
       "            continue;\n",
       "        }\n",
       "        fig.buttons[key].disabled = !msg[key];\n",
       "        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n",
       "    if (msg['mode'] === 'PAN') {\n",
       "        fig.buttons['Pan'].classList.add('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    } else if (msg['mode'] === 'ZOOM') {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.add('active');\n",
       "    } else {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message('ack', {});\n",
       "};\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function (fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = 'image/png';\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src\n",
       "                );\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data\n",
       "            );\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        } else if (\n",
       "            typeof evt.data === 'string' &&\n",
       "            evt.data.slice(0, 21) === 'data:image/png;base64'\n",
       "        ) {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig['handle_' + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\n",
       "                \"No handler for the '\" + msg_type + \"' message type: \",\n",
       "                msg\n",
       "            );\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\n",
       "                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n",
       "                    e,\n",
       "                    e.stack,\n",
       "                    msg\n",
       "                );\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "};\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function (e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e) {\n",
       "        e = window.event;\n",
       "    }\n",
       "    if (e.target) {\n",
       "        targ = e.target;\n",
       "    } else if (e.srcElement) {\n",
       "        targ = e.srcElement;\n",
       "    }\n",
       "    if (targ.nodeType === 3) {\n",
       "        // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "    }\n",
       "\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    var boundingRect = targ.getBoundingClientRect();\n",
       "    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n",
       "    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n",
       "\n",
       "    return { x: x, y: y };\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys(original) {\n",
       "    return Object.keys(original).reduce(function (obj, key) {\n",
       "        if (typeof original[key] !== 'object') {\n",
       "            obj[key] = original[key];\n",
       "        }\n",
       "        return obj;\n",
       "    }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function (event, name) {\n",
       "    var canvas_pos = mpl.findpos(event);\n",
       "\n",
       "    if (name === 'button_press') {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {\n",
       "        x: x,\n",
       "        y: y,\n",
       "        button: event.button,\n",
       "        step: event.step,\n",
       "        guiEvent: simpleKeys(event),\n",
       "    });\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function (_event, _name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.key_event = function (event, name) {\n",
       "    // Prevent repeat events\n",
       "    if (name === 'key_press') {\n",
       "        if (event.which === this._key) {\n",
       "            return;\n",
       "        } else {\n",
       "            this._key = event.which;\n",
       "        }\n",
       "    }\n",
       "    if (name === 'key_release') {\n",
       "        this._key = null;\n",
       "    }\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which !== 17) {\n",
       "        value += 'ctrl+';\n",
       "    }\n",
       "    if (event.altKey && event.which !== 18) {\n",
       "        value += 'alt+';\n",
       "    }\n",
       "    if (event.shiftKey && event.which !== 16) {\n",
       "        value += 'shift+';\n",
       "    }\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, { key: value, guiEvent: simpleKeys(event) });\n",
       "    return false;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function (name) {\n",
       "    if (name === 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message('toolbar_button', { name: name });\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function (tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Left button pans, Right button zooms\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\\nx/y fixes axis, CTRL fixes aspect\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";/* global mpl */\n",
       "\n",
       "var comm_websocket_adapter = function (comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function () {\n",
       "        comm.close();\n",
       "    };\n",
       "    ws.send = function (m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function (msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data']);\n",
       "    });\n",
       "    return ws;\n",
       "};\n",
       "\n",
       "mpl.mpl_figure_comm = function (comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = document.getElementById(id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm);\n",
       "\n",
       "    function ondownload(figure, _format) {\n",
       "        window.open(figure.canvas.toDataURL());\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy, ondownload, element);\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element;\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error('Failed to find cell for figure', id, fig);\n",
       "        return;\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function (fig, msg) {\n",
       "    var width = fig.canvas.width / mpl.ratio;\n",
       "    fig.root.removeEventListener('remove', this._remove_fig_handler);\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable();\n",
       "    fig.parent_element.innerHTML =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "    fig.close_ws(fig, msg);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.close_ws = function (fig, msg) {\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function (_remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width / mpl.ratio;\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] =\n",
       "        '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message('ack', {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () {\n",
       "        fig.push_to_output();\n",
       "    }, 1000);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function () {\n",
       "    var fig = this;\n",
       "\n",
       "    var toolbar = document.createElement('div');\n",
       "    toolbar.classList = 'btn-toolbar';\n",
       "    this.root.appendChild(toolbar);\n",
       "\n",
       "    function on_click_closure(name) {\n",
       "        return function (_event) {\n",
       "            return fig.toolbar_button_onclick(name);\n",
       "        };\n",
       "    }\n",
       "\n",
       "    function on_mouseover_closure(tooltip) {\n",
       "        return function (event) {\n",
       "            if (!event.currentTarget.disabled) {\n",
       "                return fig.toolbar_button_onmouseover(tooltip);\n",
       "            }\n",
       "        };\n",
       "    }\n",
       "\n",
       "    fig.buttons = {};\n",
       "    var buttonGroup = document.createElement('div');\n",
       "    buttonGroup.classList = 'btn-group';\n",
       "    var button;\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",
       "            /* Instead of a spacer, we start a new button group. */\n",
       "            if (buttonGroup.hasChildNodes()) {\n",
       "                toolbar.appendChild(buttonGroup);\n",
       "            }\n",
       "            buttonGroup = document.createElement('div');\n",
       "            buttonGroup.classList = 'btn-group';\n",
       "            continue;\n",
       "        }\n",
       "\n",
       "        button = fig.buttons[name] = document.createElement('button');\n",
       "        button.classList = 'btn btn-default';\n",
       "        button.href = '#';\n",
       "        button.title = name;\n",
       "        button.innerHTML = '<i class=\"fa ' + image + ' fa-lg\"></i>';\n",
       "        button.addEventListener('click', on_click_closure(method_name));\n",
       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message pull-right';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = document.createElement('div');\n",
       "    buttongrp.classList = 'btn-group inline pull-right';\n",
       "    button = document.createElement('button');\n",
       "    button.classList = 'btn btn-mini btn-primary';\n",
       "    button.href = '#';\n",
       "    button.title = 'Stop Interaction';\n",
       "    button.innerHTML = '<i class=\"fa fa-power-off icon-remove icon-large\"></i>';\n",
       "    button.addEventListener('click', function (_evt) {\n",
       "        fig.handle_close(fig, {});\n",
       "    });\n",
       "    button.addEventListener(\n",
       "        'mouseover',\n",
       "        on_mouseover_closure('Stop Interaction')\n",
       "    );\n",
       "    buttongrp.appendChild(button);\n",
       "    var titlebar = this.root.querySelector('.ui-dialog-titlebar');\n",
       "    titlebar.insertBefore(buttongrp, titlebar.firstChild);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._remove_fig_handler = function () {\n",
       "    this.close_ws(this, {});\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function (el) {\n",
       "    el.style.boxSizing = 'content-box'; // override notebook setting of border-box.\n",
       "    el.addEventListener('remove', this._remove_fig_handler);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function (el) {\n",
       "    // this is important to make the div 'focusable\n",
       "    el.setAttribute('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",
       "    } else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\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",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which === 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    fig.ondownload(fig, null);\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 < ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code') {\n",
       "            for (var j = 0; j < cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 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(\n",
       "        'matplotlib',\n",
       "        mpl.mpl_figure_comm\n",
       "    );\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\" width=\"432\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rw = RandomWalk(p, x_0, n, t, m)\n",
    "rw.generate_position_distribution()\n",
    "rw.animate_position_distribution()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "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": "3.8.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}