{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 4.3: Bootstrapping theory with... bootstrapping!\n",
    "\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say we have a data set with $n$ unique measurements. \n",
    "\n",
    "\n",
    "**a)** Show that on average a fraction of $(1 - 1/n)^n$ of the measurements do not appear in a bootstrap sample. Note that for large $n$, this is approximately  $1/\\mathrm{e} \\approx 1/2.7$, since $\\lim_{n\\to\\infty}(1 - 1/n)^n =  1/\\mathrm{e}$. This part of the problem is optional and not graded.\n",
    "\n",
    "**b)** Use a bootstrapping approach to demonstrate that this is indeed true. Hint: Think about a convenient \"data set\" to use for drawing samples. This is kind of fun; you're investigating some theory behind bootstrapping with bootstrapping!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br />"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.9.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}