102 lines
2.1 KiB
Plaintext
102 lines
2.1 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "206bf674",
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"metadata": {},
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"outputs": [],
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"source": [
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"import os\n",
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"import sys\n",
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"\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"import pandas as pd\n",
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"import seaborn as sns\n",
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"\n",
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"sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "612bd02c",
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"metadata": {},
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"source": [
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"# Chapter 4 Example Problems"
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]
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},
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{
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"cell_type": "markdown",
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"id": "5ce47b88",
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"metadata": {},
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"source": [
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"4.2 \n",
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"\n",
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"a. \n",
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"\n",
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"$\\int_{-\\infty}^{\\infty}f_X(u)du = 1$\n",
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"\n",
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"ie\n",
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"\n",
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"\\begin{align*}\n",
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"\\int_{-\\infty}^{0^+}f_X(u)du + \\int_{0}^{\\infty}f_X(x)dx = 1 \\\\\n",
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"0 + \\int_{0}^{\\infty}ce^{-x}dx = 1 \\\\\n",
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"c\\cdot\\lim_{t \\to \\infty }\\int_{0}^{t}e^{-x}dx = 1 \\\\\n",
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"c\\cdot\\lim_{t \\to \\infty } [-e^{-x}]_0^t = 1 \\\\\n",
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"c\\cdot\\lim_{t \\to \\infty } ((-e^{-t}) - (-e^{-0})) = 1 \\\\\n",
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"c\\cdot\\lim_{t \\to \\infty } (-e^{-t} + 1) = 1 \\\\\n",
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"c\\cdot 1 = 1 \\\\\n",
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"c = 1\n",
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"\\end{align*}\n",
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"\n",
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"b.\n",
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"\n",
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"Note that\n",
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"\n",
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"$$F_X(x) = \\int_{-\\infty}^x f_X(u)du$$\n",
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"\n",
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"and $c=1$\n",
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"\n",
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"so\n",
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"\n",
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"\\begin{align*}\n",
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"F_X(x) = \\begin{cases} \n",
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"1-e^{-x} & \\text{for } x \\geq 0 \\\\ \n",
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"0 & \\text{otherwise} \n",
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"\\end{cases}\n",
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"\\end{align*}\n",
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"\n",
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"c. $F_X(3) - F_X(1) = $"
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]
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},
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{
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"cell_type": "markdown",
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"id": "95a3b21c",
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"metadata": {},
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "roadmap (3.14.5)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.14.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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