Bootstrapping_example.ipynb

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   "source": [
    "using Distributions, Random"
   ]
  },
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   "cell_type": "markdown",
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   "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."
   ]
  },
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   "cell_type": "code",
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    {
     "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."
   ]
  }
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