4-1-Poisson-Numerical.ipynb

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   "source": [
    "# The Poisson Process: Section 1 - Numerical Example"
   ]
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   "cell_type": "code",
   "execution_count": 1,
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   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import ipywidgets as widgets\n",
    "from IPython.display import display, Math"
   ]
  },
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import bokeh.plotting as bplt\n",
    "from bokeh.models import Range1d, LabelSet, ColumnDataSource, Arrow, TeeHead, VeeHead, NormalHead, OpenHead, Whisker\n",
    "from bokeh.models.glyphs import VBar, Line, Step\n",
    "from bokeh.models.markers import Circle\n",
    "from bokeh.io import output_notebook, push_notebook\n",
    "output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Passengers Arriving to a Bus Stop\n",
    "\n",
    "We want to model the arrival of passengers to a bus stop (say, the Stanford Circle shuttle stop).  Our information is that passengers arrive *on average* at a rate of 2 per minute, but the actual arrival process is not deterministic: it would be unrealistic to assume that the next passenger arrives exactly 30 seconds after the previous one.\n",
    "\n",
    "How can we add stochastic variability to the arrival process?  (And do so in a way that allows tractable analysis?)\n",
    "* Time interval between successive arrivals is random (must be non-negative).\n",
    "* Independent and identically distributed.\n",
    "* On average, 30 seconds.\n",
    "\n",
    "Let $X_n$ be the time interval between the $n-1$st and $n$th passenger, so $(X_1, X_2, \\dots)$ is a sequence of independent and identically distributed random variables with $\\mathsf{E}[X_i] = 0.5$.  Note that this still allows some flexibility.  In particular, we can choose any non-negative distribution with the appropriate mean: Exponential, Uniform, Lognormal, Logistic, etc.\n",
    "\n",
    "For reasons that will become clear later, we will assume that the interarrival times are exponentially distributed.\n",
    "$$X_i \\sim \\text{Exponential}(\\lambda)$$\n",
    "with $\\lambda = 2$ for $\\mathsf{E}[X_i] = 0.5$.\n",
    "\n",
    "Let $S_n$ be the time at which the $n$th passenger arrives:\n",
    "$$ S_n \\overset{\\mathsf{def}}{=} \\sum_{i=1}^n X_i $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $N(t)$ be the number of passengers that have arrived by time $t$.\n",
    "$$N(t) \\overset{\\mathsf{def}}{=} \\max\\{n\\,:\\, S_n \\leq t\\}$$\n",
    "The stochastic process $\\{N(t),\\, t \\geq 0\\}$ is a type of “counting process.”  The random path represented by this stochastic process is usually best interpreted as counting arrivals over time — it increases by 1 each time there is an arrival.\n",
    "\n",
    "When the random variables $X_i$ are independent and identically distributed with $X_i \\sim \\text{Exponential}(\\lambda)$ then $\\{N(t),\\, t \\geq 0\\}$ is a **Poisson Process with rate $\\lambda$**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
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       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>S_n</th>\n",
       "      <th>X_n</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>n</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>1.286932</td>\n",
       "      <td>1.286932</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>2.580294</td>\n",
       "      <td>1.293363</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>2.769530</td>\n",
       "      <td>0.189235</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>4</td>\n",
       "      <td>3.499225</td>\n",
       "      <td>0.729695</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>5</td>\n",
       "      <td>4.539234</td>\n",
       "      <td>1.040010</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>6</td>\n",
       "      <td>4.685247</td>\n",
       "      <td>0.146013</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>7</td>\n",
       "      <td>4.764499</td>\n",
       "      <td>0.079253</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>8</td>\n",
       "      <td>5.098320</td>\n",
       "      <td>0.333820</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>9</td>\n",
       "      <td>5.309425</td>\n",
       "      <td>0.211105</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>10</td>\n",
       "      <td>5.442203</td>\n",
       "      <td>0.132778</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         S_n       X_n\n",
       "n                     \n",
       "1   1.286932  1.286932\n",
       "2   2.580294  1.293363\n",
       "3   2.769530  0.189235\n",
       "4   3.499225  0.729695\n",
       "5   4.539234  1.040010\n",
       "6   4.685247  0.146013\n",
       "7   4.764499  0.079253\n",
       "8   5.098320  0.333820\n",
       "9   5.309425  0.211105\n",
       "10  5.442203  0.132778"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "lam = 2\n",
    "steps = 10\n",
    "S_0 = 0\n",
    "X = np.random.exponential(1/lam, steps)\n",
    "S = pd.DataFrame(np.cumsum(X), index=pd.Index(range(1,steps+1), name='n'), columns=['S_n'])\n",
    "S['X_n'] = X\n",
    "display(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"a26ec314-1091-4c60-a84c-1d73a98031e0\" data-root-id=\"1002\"></div>\n"
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     "metadata": {},
     "output_type": "display_data"
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       "(function(root) {\n",
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{\"1b7d297e-62f1-4e1b-adb9-87065021d374\":{\"roots\":{\"references\":[{\"attributes\":{\"below\":[{\"id\":\"1013\",\"type\":\"LinearAxis\"}],\"center\":[{\"id\":\"1017\",\"type\":\"Grid\"},{\"id\":\"1022\",\"type\":\"Grid\"},{\"id\":\"1040\",\"type\":\"Arrow\"}],\"left\":[{\"id\":\"1018\",\"type\":\"LinearAxis\"}],\"plot_height\":200,\"renderers\":[{\"id\":\"1043\",\"type\":\"GlyphRenderer\"}],\"title\":{\"id\":\"1003\",\"type\":\"Title\"},\"toolbar\":{\"id\":\"1029\",\"type\":\"Toolbar\"},\"x_range\":{\"id\":\"1045\",\"type\":\"Range1d\"},\"x_scale\":{\"id\":\"1009\",\"type\":\"LinearScale\"},\"y_range\":{\"id\":\"1007\",\"type\":\"DataRange1d\"},\"y_scale\":{\"id\":\"1011\",\"type\":\"LinearScale\"}},\"id\":\"1002\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{\"source\":{\"id\":\"1042\",\"type\":\"ColumnDataSource\"}},\"id\":\"1044\",\"type\":\"CDSView\"},{\"attributes\":{},\"id\":\"1027\",\"type\":\"ResetTool\"},{\"attributes\":{\"fixed_location\":0,\"formatter\":{\"id\":\"1049\",\"type\":\"BasicTickFormatter\"},\"ticker\":{\"id\":\"1014\",\"type\":\"BasicTicker\"}},\"id\":\"1013\",\"type\":\"LinearAxis\"},{\"attributes\":{},\"id\":\"1028\",\"type\":\"HelpTool\"},{\"attributes\":{},\"id\":\"1011\",\"type\":\"LinearScale\"},{\"attributes\":{},\"id\":\"1024\",\"type\":\"WheelZoomTool\"},{\"attributes\":{\"end\":{\"id\":\"1038\",\"type\":\"VeeHead\"},\"source\":{\"id\":\"1039\",\"type\":\"ColumnDataSource\"},\"start\":{\"id\":\"1037\",\"type\":\"VeeHead\"},\"x_end\":{\"field\":\"end\"},\"x_start\":{\"field\":\"start\"},\"y_end\":{\"value\":0.5},\"y_start\":{\"value\":0.5}},\"id\":\"1040\",\"type\":\"Arrow\"},{\"attributes\":{},\"id\":\"1019\",\"type\":\"BasicTicker\"},{\"attributes\":{},\"id\":\"1014\",\"type\":\"BasicTicker\"},{\"attributes\":{\"callback\":null,\"data\":{\"end\":{\"__ndarray__\":\"Ey1gpEWX9D9gft9qcaQEQGeuR0b/JwZAuD/sb2n+C0DJ/M/9LCgSQAN32E6xvRJAUksV89gOE0Bj3trlrWQUQAZoytbZPBVAcGwz09DEFUA=\",\"dtype\":\"float64\",\"shape\":[10]},\"n\":[1,2,3,4,5,6,7,8,9,10],\"start\":{\"__ndarray__\":\"AAAAAAAAAAATLWCkRZf0P2B+32pxpARAZ65HRv8nBkC4P+xvaf4LQMn8z/0sKBJAA3fYTrG9EkBSSxXz2A4TQGPe2uWtZBRABmjK1tk8FUA=\",\"dtype\":\"float64\",\"shape\":[10]}},\"selected\":{\"id\":\"1054\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"1055\",\"type\":\"UnionRenderers\"}},\"id\":\"1039\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"data_source\":{\"id\":\"1042\",\"type\":\"ColumnDataSource\"},\"glyph\":{\"id\":\"1036\",\"type\":\"Circle\"},\"hover_glyph\":null,\"muted_glyph\":null,\"view\":{\"id\":\"1044\",\"type\":\"CDSView\"}},\"id\":\"1043\",\"type\":\"GlyphRenderer\"},{\"attributes\":{\"ticker\":{\"id\":\"1014\",\"type\":\"BasicTicker\"},\"visible\":false},\"id\":\"1017\",\"type\":\"Grid\"},{\"attributes\":{\"active_drag\":\"auto\",\"active_inspect\":\"auto\",\"active_multi\":null,\"active_scroll\":\"auto\",\"active_tap\":\"auto\",\"tools\":[{\"id\":\"1023\",\"type\":\"PanTool\"},{\"id\":\"1024\",\"type\":\"WheelZoomTool\"},{\"id\":\"1025\",\"type\":\"BoxZoomTool\"},{\"id\":\"1026\",\"type\":\"SaveTool\"},{\"id\":\"1027\",\"type\":\"ResetTool\"},{\"id\":\"1028\",\"type\":\"HelpTool\"}]},\"id\":\"1029\",\"type\":\"Toolbar\"},{\"attributes\":{\"callback\":null,\"data\":{\"S_n\":{\"__ndarray__\":\"Ey1gpEWX9D9gft9qcaQEQGeuR0b/JwZAuD/sb2n+C0DJ/M/9LCgSQAN32E6xvRJAUksV89gOE0Bj3trlrWQUQAZoytbZPBVAcGwz09DEFUA=\",\"dtype\":\"float64\",\"shape\":[10]},\"X_n\":{\"__ndarray__\":\"Ey1gpEWX9D+tz14xnbH0P3YAg7bdOMg/Q0WSpqhZ5z+0c2cX4aPwP0NHDyGKsMI/vhM1D+lJtD8XMVksT13VP2I08R1+Bcs/Q40gjd/+wD8=\",\"dtype\":\"float64\",\"shape\":[10]},\"n\":[1,2,3,4,5,6,7,8,9,10]},\"selected\":{\"id\":\"1051\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"1052\",\"type\":\"UnionRenderers\"}},\"id\":\"1042\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"dimension\":1,\"ticker\":{\"id\":\"1019\",\"type\":\"BasicTicker\"},\"visible\":false},\"id\":\"1022\",\"type\":\"Grid\"},{\"attributes\":{\"formatter\":{\"id\":\"1047\",\"type\":\"BasicTickFormatter\"},\"ticker\":{\"id\":\"1019\",\"type\":\"BasicTicker\"},\"visible\":false},\"id\":\"1018\",\"type\":\"LinearAxis\"},{\"attributes\":{},\"id\":\"1049\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{\"callback\":null,\"end\":6.442202854169508},\"id\":\"1045\",\"type\":\"Range1d\"},{\"attributes\":{},\"id\":\"1047\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{\"fill_color\":{\"value\":\"red\"},\"line_color\":{\"value\":\"red\"},\"size\":{\"units\":\"screen\",\"value\":10},\"x\":{\"field\":\"S_n\"},\"y\":{\"value\":0}},\"id\":\"1036\",\"type\":\"Circle\"},{\"attributes\":{},\"id\":\"1051\",\"type\":\"Selection\"},{\"attributes\":{\"size\":5},\"id\":\"1038\",\"type\":\"VeeHead\"},{\"attributes\":{},\"id\":\"1052\",\"type\":\"UnionRenderers\"},{\"attributes\":{\"align\":\"center\",\"text\":\"Timeline of Arrivals\",\"text_font_size\":{\"value\":\"25px\"}},\"id\":\"1003\",\"type\":\"Title\"},{\"attributes\":{\"size\":5},\"id\":\"1037\",\"type\":\"VeeHead\"},{\"attributes\":{\"bottom_units\":\"screen\",\"fill_alpha\":{\"value\":0.5},\"fill_color\":{\"value\":\"lightgrey\"},\"left_units\":\"screen\",\"level\":\"overlay\",\"line_alpha\":{\"value\":1.0},\"line_color\":{\"value\":\"black\"},\"line_dash\":[4,4],\"line_width\":{\"value\":2},\"render_mode\":\"css\",\"right_units\":\"screen\",\"top_units\":\"screen\"},\"id\":\"1053\",\"type\":\"BoxAnnotation\"},{\"attributes\":{},\"id\":\"1023\",\"type\":\"PanTool\"},{\"attributes\":{},\"id\":\"1054\",\"type\":\"Selection\"},{\"attributes\":{\"callback\":null},\"id\":\"1007\",\"type\":\"DataRange1d\"},{\"attributes\":{\"overlay\":{\"id\":\"1053\",\"type\":\"BoxAnnotation\"}},\"id\":\"1025\",\"type\":\"BoxZoomTool\"},{\"attributes\":{},\"id\":\"1055\",\"type\":\"UnionRenderers\"},{\"attributes\":{},\"id\":\"1009\",\"type\":\"LinearScale\"},{\"attributes\":{},\"id\":\"1026\",\"type\":\"SaveTool\"}],\"root_ids\":[\"1002\"]},\"title\":\"Bokeh 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       "  var render_items = [{\"docid\":\"1b7d297e-62f1-4e1b-adb9-87065021d374\",\"roots\":{\"1002\":\"a26ec314-1091-4c60-a84c-1d73a98031e0\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt = bplt.figure(title='Timeline of Arrivals', plot_height=200)\n",
    "plt.title.align = 'center'\n",
    "plt.title.text_font_size = '25px'\n",
    "glyph_S = Circle(x=\"S_n\", y=0, size=10, line_color=\"red\", fill_color=\"red\")\n",
    "interarr = pd.DataFrame({'start': np.append(0, S['S_n'][:-1].values), 'end': S['S_n'].values}, index=pd.Index(range(1, steps+1), name='n'))\n",
    "plt.add_layout(Arrow(start=VeeHead(size=5), end=VeeHead(size=5), \n",
    "                     x_start='start', x_end='end', y_start=0.5, y_end=0.5, \n",
    "                     source=ColumnDataSource(interarr)))\n",
    "plt.add_glyph(ColumnDataSource(S), glyph_S)\n",
    "plt.grid.visible = False\n",
    "plt.yaxis.visible = False\n",
    "plt.x_range = Range1d(0, max(S['S_n'])+1)\n",
    "plt.xaxis.fixed_location = 0\n",
    "bplt.show(plt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = pd.DataFrame(np.arange(0,steps+1), \n",
    "                 index=pd.Index(np.append(np.array(S_0), S['S_n'].values), name='t'), \n",
    "                 columns=['N(t)'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "  <div class=\"bk-root\" id=\"04f51943-23e2-4985-b2e2-f8e16323e5ed\" data-root-id=\"1116\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = {\"2c41ad1f-64d1-4a1b-88e1-dd566f569923\":{\"roots\":{\"references\":[{\"attributes\":{\"below\":[{\"id\":\"1127\",\"type\":\"LinearAxis\"}],\"center\":[{\"id\":\"1131\",\"type\":\"Grid\"},{\"id\":\"1136\",\"type\":\"Grid\"}],\"left\":[{\"id\":\"1132\",\"type\":\"LinearAxis\"}],\"renderers\":[{\"id\":\"1153\",\"type\":\"GlyphRenderer\"},{\"id\":\"1156\",\"type\":\"GlyphRenderer\"}],\"title\":{\"id\":\"1117\",\"type\":\"Title\"},\"toolbar\":{\"id\":\"1143\",\"type\":\"Toolbar\"},\"x_range\":{\"id\":\"1158\",\"type\":\"Range1d\"},\"x_scale\":{\"id\":\"1123\",\"type\":\"LinearScale\"},\"y_range\":{\"id\":\"1121\",\"type\":\"DataRange1d\"},\"y_scale\":{\"id\":\"1125\",\"type\":\"LinearScale\"}},\"id\":\"1116\",\"subtype\":\"Figure\",\"type\":\"Plot\"},{\"attributes\":{\"align\":\"center\",\"text\":\"Counting Process {N(t), t \\u2265 0}\",\"text_font_size\":{\"value\":\"25px\"}},\"id\":\"1117\",\"type\":\"Title\"},{\"attributes\":{},\"id\":\"1170\",\"type\":\"BasicTickFormatter\"},{\"attributes\":{\"callback\":null,\"data\":{\"N(t)\":[0,1,2,3,4,5,6,7,8,9,10],\"t\":{\"__ndarray__\":\"AAAAAAAAAAATLWCkRZf0P2B+32pxpARAZ65HRv8nBkC4P+xvaf4LQMn8z/0sKBJAA3fYTrG9EkBSSxXz2A4TQGPe2uWtZBRABmjK1tk8FUBwbDPT0MQVQA==\",\"dtype\":\"float64\",\"shape\":[11]}},\"selected\":{\"id\":\"1174\",\"type\":\"Selection\"},\"selection_policy\":{\"id\":\"1175\",\"type\":\"UnionRenderers\"}},\"id\":\"1150\",\"type\":\"ColumnDataSource\"},{\"attributes\":{\"bottom_units\":\"screen\",\"fill_alpha\":{\"value\":0.5},\"fill_color\":{\"value\":\"lightgrey\"},\"left_units\":\"screen\",\"level\":\"overlay\",\"line_alpha\":{\"value\":1.0},\"line_color\":{\"value\":\"black\"},\"line_dash\":[4,4],\"line_width\":{\"value\":2},\"render_mode\":\"css\",\"right_units\":\"screen\",\"top_units\":\"screen\"},\"id\":\"1178\",\"type\":\"BoxAnnotation\"},{\"attributes\":{},\"id\":\"1142\",\"type\":\"HelpTool\"},{\"attributes\":{},\"id\":\"1176\",\"type\":\"Selection\"},{\"attributes\":{\"axis_label\":\"# 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Application\",\"version\":\"1.3.4\"}};\n",
       "  var render_items = [{\"docid\":\"2c41ad1f-64d1-4a1b-88e1-dd566f569923\",\"roots\":{\"1116\":\"04f51943-23e2-4985-b2e2-f8e16323e5ed\"}}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "1116"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt = bplt.figure(title='Counting Process {N(t), t ≥ 0}')\n",
    "plt.title.align = 'center'\n",
    "plt.title.text_font_size = '25px'\n",
    "source = ColumnDataSource(N)\n",
    "glyph_N = Step(x=\"t\", y=\"N(t)\", line_color=\"blue\", mode=\"after\")\n",
    "glyph_S = Circle(x=\"t\", y=0, size=10, line_color=\"red\", fill_color=\"red\")\n",
    "plt.add_glyph(source, glyph_N)\n",
    "plt.add_glyph(ColumnDataSource(N.iloc[1:]), glyph_S)\n",
    "plt.xaxis.axis_label = \"time\"\n",
    "plt.yaxis.axis_label = \"# of arrivals\"\n",
    "#plt.y_range = Range1d(0, max(N['N(t)'])+1)\n",
    "plt.x_range = Range1d(0, max(S['S_n'])+1)\n",
    "bplt.show(plt)"
   ]
  }
 ],
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