Added variance to lecture 2.

hw/hw_01.ipynb

@@ -67,8 +67,18 @@
     "p(x_1 \\le X \\le x_2|I) = F(x_2) - F(x_1).\n",
     "$$\n",
     "\n",
-    "3. **Note**: There is no need to remember anything of what follows. The purpose of this exercise is to get familiar with the laws of probability. \n",
-    "Let $X,Y,Z$, and $W$ be random variables. "
+    "4. Let $X$ be a random variable. Prove that:\n",
+    "$$\n",
+    "\\mathbb{V}[X] = \\mathbb{E}[X^2] - \\left(\\mathbb{E}[X]\\right)^2.\n",
+    "$$\n",
+    "\n",
+    "Let $X$ be the following random variable:\n",
+    "$$\n",
+    "X = \\begin{cases}\n",
+    "1,& \\theta,\\\\\n",
+    "0,& 1-\\theta,\n",
+    "\\end{cases}\n",
+    "$$"
    ]
   },
   {

lectures/lec_02.ipynb

@@ -544,6 +544,31 @@
     "\\mathbb{E}[X|Y=y] := \\mathbb{E}[X|Y=y,I] = \\int xp(x|y)dx.\n",
     "$$"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Variance of a Random Variable\n",
+    "\n",
+    "The variance of $X$ is defined to be:\n",
+    "$$\n",
+    "\\mathbb{V}[X] = \\int (x - \\mathbb{E})^2p(x)dx.\n",
+    "$$\n",
+    "It is easy to prove (and a very useful formulat to remember), that:\n",
+    "$$\n",
+    "\\mathbb{V}[X] = \\mathbb{E}[X^2] - \\left(\\mathbb{E}[X]\\right)^2.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {

lectures/lec_03.ipynb

@@ -11,6 +11,37 @@
     "+ Learn about some famous continous PDFs"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Discrete random variables\n",
+    "\n",
+    "### Coin flipping: Heads = 1, Tails = 0\n",
+    "\n",
+    "Let $X$ be the following random variable:\n",
+    "$$\n",
+    "X = \\begin{cases}\n",
+    "1,& \\theta,\\\\\n",
+    "0,& 1-\\theta,\n",
+    "\\end{cases}\n",
+    "$$\n",
+    "where $\\theta\\in[0,1]$ is the probability of getting heads (i.e., 1).\n",
+    "The PDF of this random variable has two values:\n",
+    "$$\n",
+    "p(X=1|\\theta) = \\theta,\n",
+    "$$\n",
+    "and\n",
+    "$$\n",
+    "p(X=0|\\theta) = 1-\\theta.\n",
+    "$$\n",
+    "\n",
+    "The expected value is:\n",
+    "$$\n",
+    "\\mathbb{E}[X|\\theta] = \\sum_x xp(X=x|\\theta) = 1\\times p(X=1|\\theta) + 0\\times p(X=0|\\theta) = \\theta.\n",
+    "$$"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -424,7 +455,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython2",
-   "version": "2.7.10"
+   "version": "2.7.11"
   },
   "latex_envs": {
    "bibliofile": "biblio.bib",