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  {
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   "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": "a6732353-51d5-4478-9cf8-5834e57e5a4e",
   "metadata": {},
   "source": [
    "# Chapter 4 Notes"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f0046c2",
   "metadata": {},
   "source": [
    "## Continuous Random Variables and their Distributions (4.1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea9b1f96",
   "metadata": {},
   "source": [
    "***Definition*** A random variable $X$ with CDF $F_X(x)$ is said to be continuous if $F_X(x)$ is a continuous for all $x \\in \\mathbb{R}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a96da4d3",
   "metadata": {},
   "source": [
    "## Probability Density Function (PDF) (4.1.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c62129a5",
   "metadata": {},
   "source": [
    "***Definition*** Consider a continuous random variable $X$ with an absolutely continuous CDF $F_X(x)$. The function $f_X(x)$ defined by\n",
    "\n",
    "$$f_X(x) = \\frac{dF_X(x)}{dx} = F'_X(x) \\quad \\text{if } F_X(x) \\text{ is differentiable at } x$$\n",
    " \n",
    "is called the probability density function (PDF) of $X$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df411869",
   "metadata": {},
   "source": [
    "NOTE: The PDF being constant implies uniformity\n",
    "\n",
    "NOTE: For small values of $\\delta$,\n",
    "\n",
    "$$P(x \\lt X \\leq x + \\delta) \\approx  f_X(x)\\delta$$\n",
    "\n",
    "Thus if $f_X(x_1) \\gt f_X(x_2)$, we can say $P(x_1 \\lt X \\leq x_1 + \\delta) \\gt P(x_2 \\lt X \\leq x_2 + \\delta)$, ie the value of $X$ is more likely to be around $x_1$ then $x_2$\n",
    "\n",
    "NOTE: The CDF can be obtained from the PDF via (assuming absolute continuity)\n",
    "\n",
    "$$F_X(x) = \\int_{-\\infty}^x f_X(u)du$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f29a3bfb",
   "metadata": {},
   "source": [
    "***Properties*** Consider a continuous random variable $X$ with PDF $f_X(x)$. We have\n",
    "- $f_X(x) \\geq 0, \\forall x \\in \\mathbb{R}$\n",
    "- $\\int_{-\\infty}^{\\infty}f_X(u)du = 1$\n",
    "- $P(a \\lt X \\leq b) = F_X(b) - F_X(a) = \\int_a^bf_X(u)du$\n",
    "- For a set $A$, $P(X \\in A) = \\int_Af_X(u)du$. However, set $A$ must satisfy:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f546fed",
   "metadata": {},
   "source": [
    "***Definition*** If $X$ is a continuous random variable, we can write the range of $X$ as\n",
    "\n",
    "$$R_X = \\{ x \\mid f_X(x) \\gt 0 \\}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "170db3a0",
   "metadata": {},
   "source": [
    "***Property*** The expected value if a continuous random variable $X$ is\n",
    "\n",
    "$$E[X] = \\int_{-\\infty}^{\\infty}xf_X(x)dx$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "deccea51",
   "metadata": {},
   "source": [
    "***Property*** Law of the unconscious statistician (LOTUS) for continuous random variables\n",
    "\n",
    "$$E[g(X)] = \\int_{-\\infty}^{\\infty}g(x)f_X(x)dx$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c0a20195",
   "metadata": {},
   "source": [
    "***Property*** Variance for a continuous random variable $X$, we can write\n",
    "\n",
    "\\begin{align*}\n",
    "\\text{Var}(X) &= E[(X - E[x])^2] = \\int_{-\\infty}^{\\infty}(x - E[X])^2f_X(x)dx \\\\\n",
    "&= E[X^2] - E[X]^2 = \\int_{-\\infty}^{\\infty}x^2f_X(x)dx - E[X]^2\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f2f6b72",
   "metadata": {},
   "source": [
    "## Functions of Continuous Random Variables"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4a95738",
   "metadata": {},
   "source": [
    "***Theorem***"
   ]
  }
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