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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": "612bd02c",
   "metadata": {},
   "source": [
    "# Chapter 4 Example Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ce47b88",
   "metadata": {},
   "source": [
    "4.2 \n",
    "\n",
    "a. \n",
    "\n",
    "$\\int_{-\\infty}^{\\infty}f_X(u)du = 1$\n",
    "\n",
    "ie\n",
    "\n",
    "\\begin{align*}\n",
    "\\int_{-\\infty}^{0^+}f_X(u)du + \\int_{0}^{\\infty}f_X(x)dx = 1 \\\\\n",
    "0 + \\int_{0}^{\\infty}ce^{-x}dx = 1 \\\\\n",
    "c\\cdot\\lim_{t \\to \\infty }\\int_{0}^{t}e^{-x}dx = 1 \\\\\n",
    "c\\cdot\\lim_{t \\to \\infty } [-e^{-x}]_0^t = 1 \\\\\n",
    "c\\cdot\\lim_{t \\to \\infty } ((-e^{-t}) - (-e^{-0})) = 1 \\\\\n",
    "c\\cdot\\lim_{t \\to \\infty } (-e^{-t} + 1) = 1 \\\\\n",
    "c\\cdot 1 = 1 \\\\\n",
    "c = 1\n",
    "\\end{align*}\n",
    "\n",
    "b.\n",
    "\n",
    "Note that\n",
    "\n",
    "$$F_X(x) = \\int_{-\\infty}^x f_X(u)du$$\n",
    "\n",
    "and $c=1$\n",
    "\n",
    "so\n",
    "\n",
    "\\begin{align*}\n",
    "F_X(x) = \\begin{cases} \n",
    "1-e^{-x}     & \\text{for } x \\geq 0 \\\\  \n",
    "0     & \\text{otherwise} \n",
    "\\end{cases}\n",
    "\\end{align*}\n",
    "\n",
    "c. $F_X(3) - F_X(1) = $"
   ]
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   "id": "95a3b21c",
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   "source": []
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