Chapter 3 progress

study/001_introduction-to-probability-statistics-and-random-processes/README.md

@@ -4,7 +4,7 @@
 
 total pages=1007
 
-**Currently reading:** chapter 3, page 157
+**Currently reading:** chapter 3, page 160
 
 TODO:
 

study/001_introduction-to-probability-statistics-and-random-processes/ch2/example-problems.ipynb

@@ -1,57 +0,0 @@
-{
- "cells": [
-  {
-   "cell_type": "code",
-   "execution_count": 1,
-   "id": "206bf674",
-   "metadata": {},
-   "outputs": [],
-   "source": [
-    "import os\n",
-    "import sys\n",
-    "\n",
-    "import matplotlib.pyplot as plt\n",
-    "import numpy as np\n",
-    "import pandas as pd\n",
-    "import seaborn as sns\n",
-    "\n",
-    "sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "612bd02c",
-   "metadata": {},
-   "source": [
-    "# Chapter 2 Example Problems"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "id": "b7f23a4f",
-   "metadata": {},
-   "source": []
-  }
- ],
- "metadata": {
-  "kernelspec": {
-   "display_name": "roadmap (3.14.5)",
-   "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": "3.14.5"
-  }
- },
- "nbformat": 4,
- "nbformat_minor": 5
-}

study/001_introduction-to-probability-statistics-and-random-processes/ch3/notes.ipynb

@@ -31,14 +31,75 @@
    "id": "9f0046c2",
    "metadata": {},
    "source": [
-    "## Random Variables (3.1.1 - 3.1.2)\n",
+    "## Random Variables (3.1.1 - 3.1.2)"
-    "\n",
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "7c73b89f",
+   "metadata": {},
+   "source": [
     "***Definition.*** Random Variables: \\\n",
     "A random variable $X$ is a function from the sample space to the real numbers. ie\n",
     "$$X : S \\to \\mathbb{R}$$\n",
     "\n",
-    "***Definition.*** $X$ is a discrete random variable, if its range is countable\n"
+    "Each outcome ($\\omega$) in the sample space must have a $X(\\omega)$ defined. \n",
+    "\n",
+    "> Note that $X$ is a deterministic function. The randomness comes from the fact that we dont know the inputs to $X$, ie the outcome of the random experiment"
    ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "afa12aa6",
+   "metadata": {},
+   "source": [
+    "***Definition.*** $X$ is a discrete random variable, if its range is countable"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "abd4a3ea",
+   "metadata": {},
+   "source": [
+    "### More on $X$\n",
+    "\n",
+    "Let $\\Omega$ be a sample space and let $X$ be a random variable on $\\Omega$\n",
+    "\n",
+    "- $X$ is a function, and $\\forall \\omega \\in \\Omega, X(\\omega) \\in \\mathbb{R}\\quad$ (ie $X(\\omega)$ is defined on all **outcomes**) \n",
+    "- $P_x(1)$ is asking: For event $A = \\{\\omega \\in \\Omega \\mid X(\\omega) = 1 \\}$, what is $P(A)$?\n",
+    "- $X$ induces a partition of $\\Omega$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "a7dab8e8",
+   "metadata": {},
+   "source": [
+    "***Definition.*** Let $X$ be a discrete random variable with range $R_X = \\{x_1, x_2, x_3, \\dots\\}$ (finite or countably infinite). The function\n",
+    "\n",
+    "$$P_X(x_k) = P(X = x_k), \\text{for} k = 1,2,3,\\dots,$$\n",
+    "\n",
+    "is called the *probability mass function (PMF)* of $X$. (also called the probability distribution)\n",
+    "\n",
+    "Note that if $x \\notin R_X$, then $P_X(x) = 0$ "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84ca67da",
+   "metadata": {},
+   "source": [
+    "## Properties of PMF\n",
+    "- $0 \\geq P_X(x) \\geq 1, \\forall x$\n",
+    "- $\\sum_{x \\in R_X}P_X(x) = 1$ \n",
+    "- for any set $A \\subset R_X, P(X \\in A) = \\sum_{x \\in A} P_X(x)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "57323d97",
+   "metadata": {},
+   "source": []
   }
  ],
  "metadata": {