Poisson.ipynb

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    "import numpy as np\n",
    "from matplotlib import pyplot as plt\n",
    "import pandas as pd\n",
    "from scipy import stats\n",
    "import statsmodels.api as sm\n",
    "\n",
    "import matplotlib\n",
    "matplotlib.use('nbagg')\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Poisson distribution"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Discrete distribution that models event counts or arrivals with a fixed rate $\\mu$\n",
    "\n",
    "* Arrivals independent of each other"
   ]
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   "cell_type": "markdown",
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   "source": [
    "## Probability mass function\n",
    "\n",
    "Let $X$ be the random variable that is the number of counts or arrivals\n",
    "\n",
    "\\begin{equation}\n",
    "    f_X(k; \\mu) = e^{-\\mu} \\frac{\\mu^k}{k!} \\hspace{2cm} k \\in \\{0,1,2,\\dots\\}\n",
    "    \\label{eq:pmf}\n",
    "\\end{equation}\n",
    "\n",
    "Expectation and variance: \n",
    "\n",
    "Observe that\n",
    "\n",
    "$$\n",
    "\\sum_{k=0}^{\\infty} k \\frac{\\mu^k}{k!} \n",
    "= \\sum_{k=1}^{\\infty} k \\frac{ \\mu \\mu^{k-1}}{k (k-1)!} \n",
    "= \\mu \\sum_{k=1}^{\\infty} \\frac{\\mu^{k-1}}{(k-1)!} = \\mu \\sum_{j=0}^{\\infty} \\frac{ \\mu^{j}}{j!} = \\mu e^{\\mu}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "\\sum_{k=0}^{\\infty} k^2 \\frac{\\mu^k}{k!} \n",
    " = \\mu \\sum_{k=1}^{\\infty} k \\frac{\\mu^{k-1}}{(k-1)!}\n",
    " = \\mu \\sum_{j=0}^{\\infty} (j+1) \\frac{ \\mu^{j}}{j!}\n",
    " = \\mu \\left (\\sum_{j=0}^{\\infty} j \\frac{ \\mu^{j}}{j!} + \\sum_{j=0}^{\\infty}\\frac{ \\mu^{j}}{j!} \\right )\n",
    " = \\mu (\\mu e^\\mu + e^\\mu)\n",
    " = e^{\\mu}(\\mu^2 + \\mu)\n",
    "$$\n",
    "\n",
    "We get\n",
    "\n",
    "$$ \n",
    "\\mu_X = e^{-\\mu} \\sum_{k=0}^{\\infty} k \\frac{\\mu^k}{k!} = e^{-\\mu} \\mu e^{\\mu} = \\mu\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\sigma^2_X = \\left [e^{-\\mu} \\sum_{k=0}^{\\infty} k^2 \\frac{\\mu^k}{k!} \\right ] - \\mu_X^2\n",
    "           = \\left [e^{-\\mu} e^{\\mu}(\\mu^2 + \\mu)\\right ] - \\mu^2\n",
    "           = \\mu^2 + \\mu -\\mu^2 = \\mu\n",
    "$$\n",
    "\n",
    "Moment generating function:\n",
    "$$\n",
    "M_X(t) = E(e^{tX}) = e^{-\\lambda}\\sum_{k=0}^{\\infty} e^{tk} \\frac{\\lambda^k}{k!} = e^{-\\lambda}\\sum_{k=0}^{\\infty} \\frac{(\\lambda e^t)^k}{k!} = e^{-\\lambda} e^{\\lambda e^t} = e^{\\lambda(e^t-1)}\n",
    "$$"
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       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\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",
       "    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",
       "    {\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, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\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",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\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,\n",
       "                             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\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"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\";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 = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\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.get(0);\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",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\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).html('<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'] = '<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 () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>')\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) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></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 = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.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<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('matplotlib', mpl.mpl_figure_comm);\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": [
    "# Poisson \n",
    "\n",
    "fig = plt.figure\n",
    "\n",
    "k = np.arange(16)        # Number of events: {0,1,2 ...}\n",
    "\n",
    "# We want to plot multiple lines in the same plot, so we call plot repeatedly in a loop.\n",
    "# The function returns objects whose text representation becomes output. To suppress that,\n",
    "# we assign each call to a temp variable (the _ character is a valid variable name).\n",
    "\n",
    "for mu in np.arange(1,7):\n",
    "    _ = plt.plot(k,stats.poisson.pmf(k,mu), '-.',label=r\"$\\mu=$\"+str(mu)) \n",
    "    _ = plt.legend()\n",
    "\n",
    "\n",
    "# 'set' used for setting multiple things in one call. Works only with gca(): get current axes.\n",
    "\n",
    "_ = plt.gca().set(ylabel = \"Probability\", title=\"Poisson Distribution\",xticks=k)\n",
    "\n",
    "# Notes:\n",
    "    # The Poission distribution is discrete. Lines are simply guides to the eye.\n",
    "    # The peak of the pmf plot shifts towards the value of mu (average number of events) \n",
    "    # because the modes are mu and mu-1\n",
    "                                                                       "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivation for arrivals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Let $\\lambda$ be the probability per unit time of one arrival. Consider a small interval $\\delta t$, during which the probability of more than one arrival is assumed to be zero. Let $P(n,t)$ be the probability of $n$ arrivals by time $t$. Consider the probability of $n$ arrivals by time $t+\\delta t$. This is the sum of two probabilities: \n",
    "1. the probability that there are $n-1$ arrivals by time $t$ and one arrival in the interval $\\delta t$\n",
    "2. the probability that there are $n$ arrivals by time $t$ and no arrivals during $\\delta t$\n",
    " \n",
    "\\begin{equation}\n",
    "  P(n,t+\\delta t) = P(n-1,t) \\lambda \\delta t + P(n,t)(1-\\lambda\\delta t) \n",
    " \\label{eq:Recur}\n",
    "\\end{equation}\n",
    "\n",
    "Rearranging equation (\\ref{eq:Recur}) and taking the limit as $\\delta t \\to 0$\n",
    "\n",
    "\\begin{equation}\n",
    "\\lim_{\\delta t \\to 0} \\frac{P(n,t+\\delta t)-P(n,t)}{\\delta t} = \\frac{d P(n,t)}{dt} = \\lambda [P(n-1,t) - P(n,t)]\n",
    "\\end{equation}\n",
    "Or,\n",
    "\\begin{equation}\n",
    "\\frac{d P(n,t)}{dt} + \\lambda P(n,t) = \\lambda P(n-1,t)\n",
    "\\end{equation}\n",
    "\n",
    "After multiplying with $e^{\\lambda t}$ this can be intergrated to give\n",
    "\\begin{equation}\n",
    " e^{\\lambda t} P(n,t) = \\lambda \\int_0^t e^{\\lambda t'} P(n-1,t')\\ dt'\n",
    " \\label{eq:int_sol}\n",
    "\\end{equation}\n",
    "\n",
    "This is a recurrence relation, so we have to start with $P(0,t)$. Thinking about the reasoning behind equation (\\ref{eq:Recur}), we see that\n",
    "\n",
    "$$ P(0,t+\\delta t) = P(0,t)(1-\\lambda\\delta t)$$\n",
    "which implies\n",
    "$$\\frac{d P(0,t)}{dt}  = -\\lambda P(0,t)$$\n",
    "\n",
    "This has the solution\n",
    "\\begin{equation}\n",
    " P(0,t) = e^{-\\lambda t}\n",
    "\\end{equation}\n",
    "\n",
    "Substituting this into equation (\\ref{eq:int_sol}) with $n=1$, we get\n",
    "\n",
    "$$P(1,t) = e^{-\\lambda t} (\\lambda t)$$\n",
    "\n",
    "Similarly, by successive substitution, we can obtain\n",
    "\n",
    "$$ P(2,t) = e^{-\\lambda t} \\frac{(\\lambda t)^2}{2}, \\hspace{0.5cm} P(3,t) = e^{-\\lambda t} \\frac{(\\lambda t)^3}{3\\times2},  \\hspace{0.5cm} P(4,t) = e^{-\\lambda t} \\frac{(\\lambda t)^4}{4\\times3\\times2}$$ etc.\n",
    "\n",
    "The general formula, which can be proved by induction, is\n",
    "\n",
    "\\begin{equation}\n",
    "\\boxed{\n",
    " P(n,t) = e^{-\\lambda t} \\frac{(\\lambda t)^n}{n!} \n",
    " }\n",
    "\\end{equation}\n",
    "\n",
    "The expectation and variance are $$ \\mu_X = \\lambda t, \\hspace{0.5 cm}\\sigma^2_X = \\lambda t$$\n"
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    "## Exponential distribution of inter-arrival times\n",
    " \n",
    "Let $T_n$ be the $n$th arrival time. Let $\\tau  \\equiv T_{n+1} - T_n$ be the interval until the next arrival. Then, \n",
    "\n",
    " $$\n",
    " \\begin{split}\n",
    " P[\\tau \\le t]  & = 1 - P[\\tau > t] \\\\\n",
    "     & = 1 - \\textrm{Probability of zero arrivals in the interval} \\\\\n",
    "     & = 1 - P(0,t) \\\\\n",
    "     & = 1 - e^{-\\lambda t}\n",
    " \\end{split}\n",
    "$$\n",
    "\n",
    "This is the cumulative distribution function which can be differentiated to get us the probability density function\n",
    "\n",
    "$$\n",
    " f_\\tau(t;\\lambda) = \\lambda e^{-\\lambda t}\n",
    "$$\n",
    "\n",
    "This is the **exponential distribution**, with expectation and variance\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "\\mu_\\tau & = \\lambda \\int_0^\\infty t e^{-\\lambda t} dt = \\frac{1}{\\lambda} \\\\\n",
    "\\sigma^2_\\tau & = \\left (\\lambda \\int_0^\\infty t^2 e^{-\\lambda t} dt \\right )  - \\mu^2_\\tau\n",
    "    = \\frac{2}{\\lambda^2} - \\frac{1}{\\lambda^2} = \\frac{1}{\\lambda^2}\n",
    "\\end{split}\n",
    "$$"
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