addressing issue CamDavidsonPilon#49

Author: CamDavidsonPilon

Chapter4_TheGreatestTheoremNeverTold/LawOfLargeNumbers.ipynb

@@ -220,13 +220,38 @@
       "              0 &  else\n",
       "\\end{cases}\n",
       "$$\n",
-      "Then, by the law of large numbers:\n",
+      "Then, by the law of large numbers, if we have many samples $X_i$, we can estimate the probability of an event $A$, denoted $P(A)$, by:\n",
       "\n",
-      "$$ \\frac{1}{N} \\sum_{i=1}^N \\mathbb{1}_A(X_i)$$\n",
+      "$$ \\frac{1}{N} \\sum_{i=1}^N \\mathbb{1}_A(X_i) \\rightarrow E[\\mathbb{1}_A(X)] =  P(A) $$\n",
       "\n",
-      "Again, this is fairly obvious after a moments thought: the indicator function is only 1 if the event occurs, so we are summing only the times the event occurs and dividing by the total number of trials  (consider how we usually approximate probablities using frequencies). \n"
+      "Again, this is fairly obvious after a moments thought: the indicator function is only 1 if the event occurs, so we are summing only the times the event occurs and dividing by the total number of trials  (consider how we usually approximate probablities using frequencies). For example, suppose we wish to estimate the probability that a $Z \\sim Exp(.5)$ is greater than 10, and we have many samples from a $Exp(.5)$ distribution. \n",
+      "\n",
+      "\n",
+      "$$ P( Z > 10 ) =  \\sum_{i=1}^N \\mathbb{1}_{z > 10 }(Z_i) $$\n"
      ]
     },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import pymc as mc\n",
+      "\n",
+      "N = 10000\n",
+      "print np.mean( [ mc.rexponential( 0.5 )>10 for i in range(N) ] )"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "0.0058\n"
+       ]
+      }
+     ],
+     "prompt_number": 4
+    },
     {
      "cell_type": "markdown",
      "metadata": {},
@@ -1081,7 +1106,7 @@
         "    div.text_cell_render{\n",
         "        font-family: Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
         "        line-height: 145%;\n",
-        "        font-size: 120%;\n",
+        "        font-size: 130%;\n",
         "        width:800px;\n",
         "        margin-left:auto;\n",
         "        margin-right:auto;\n",
@@ -1124,13 +1149,13 @@
         "</script>"
        ],
        "output_type": "pyout",
-       "prompt_number": 87,
+       "prompt_number": 5,
        "text": [
-        "<IPython.core.display.HTML at 0x15b2d3c8>"
+        "<IPython.core.display.HTML at 0x8b696a0>"
        ]
       }
      ],
-     "prompt_number": 87
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",

Chapter6_Priorities/Priors.ipynb

@@ -1571,7 +1571,7 @@
         "    div.text_cell_render{\n",
         "        font-family: Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
         "        line-height: 145%;\n",
-        "        font-size: 120%;\n",
+        "        font-size: 130%;\n",
         "        width:800px;\n",
         "        margin-left:auto;\n",
         "        margin-right:auto;\n",
@@ -1614,13 +1614,13 @@
         "</script>"
        ],
        "output_type": "pyout",
-       "prompt_number": 2,
+       "prompt_number": 105,
        "text": [
-        "<IPython.core.display.HTML at 0x835b6a0>"
+        "<IPython.core.display.HTML at 0x18bc92b0>"
        ]
       }
      ],
-     "prompt_number": 2
+     "prompt_number": 105
     },
     {
      "cell_type": "code",

styles/custom.css

@@ -18,7 +18,7 @@
     div.text_cell_render{
         font-family: Computer Modern, "Helvetica Neue", Arial, Helvetica, Geneva, sans-serif;
         line-height: 145%;
-        font-size: 120%;
+        font-size: 130%;
         width:800px;
         margin-left:auto;
         margin-right:auto;