Simulation_example

Simulation_example.ipynb

{
 "cells": [
  {
   "cell_type": "code",
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   "source": [
    "using Distributions, Random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we have a distribution $\\nu = \\mathrm{Unif}([0,1])$. Let's draw 10 observations from the distribution and compute the mean."
   ]
  },
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   "cell_type": "code",
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       " 0.6094400915124818\n",
       " 0.17083209035029978\n",
       " 0.0848438283876205\n",
       " 0.19735306058651214\n",
       " 0.7426141868026233\n",
       " 0.3776925769994388\n",
       " 0.007934450314826202\n",
       " 0.6488288212060249\n",
       " 0.18491729829383674\n",
       " 0.7119689348932787"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "observations = rand(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.3736425339346942"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean(observations)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From our knowledge of the distribution, we know that the true underlying mean of the distribution is 0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Draw 10 observations for just one time does not give us a good enough approximation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can repeat the above process for 1000000 times and see if the simulated mean is close to the true mean of the distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000015089902228"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simulation_mean = mean([mean(rand(10)) for i in 1:1000000])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that the simulated mean for 1000000 times with 10 observations each time get us very close to the true underlying mean of the distribution."
   ]
  }
 ],
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