{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["# Homework 7.4: Bootstrapping theory with... bootstrapping! (15 pts)\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": ["<hr>"]}], "metadata": {"kernelspec": {"display_name": "Python 3", "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": "3.7.5"}}, "nbformat": 4, "nbformat_minor": 4}