Skip to content

Instantly share code, notes, and snippets.

Embed
What would you like to do?
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"using Distributions, Random"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose we have a distribution $\\nu = \\mathrm{Unif}([0,1])$. We want to find out the standard error of the sample mean of 10 observations from this distribution with 1/10 mass on each of the observations."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"10-element Array{Float64,1}:\n",
" 0.7684476751965699\n",
" 0.940515000715187\n",
" 0.6739586945680673\n",
" 0.3954531123351086\n",
" 0.3132439558075186\n",
" 0.6625548164736534\n",
" 0.5860221243068029\n",
" 0.05213316316865657\n",
" 0.26863956854495097\n",
" 0.10887074134844155"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Random.seed!(123)\n",
"observations = rand(10)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using our knowledge of standard error of sample mean above, we can calculate the standard error of the sample mean. We have $\\sqrt{\\frac{10(1/12)}{10^2}} = \\sqrt{1/120}$ "
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.09128709291752768"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"true_std = sqrt(1/120)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.08818705818316087"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bootstrapping_std = std([mean(sample(observations, 10)) for i in 1:2000000])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notice that in this case, the standard error of the sample mean we get is far from the true standard error. This is because we are estimation $T(\\hat{\\nu})$ instead of $T(\\nu)$. These two values might not be close to each other."
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.5.1",
"language": "julia",
"name": "julia-1.5"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.5.1"
}
},
"nbformat": 4,
"nbformat_minor": 4
}
Sign up for free to join this conversation on GitHub. Already have an account? Sign in to comment
You can’t perform that action at this time.