{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "pycharm": {
     "name": "#%% md\n"
    }
   },
   "source": [
    "## Simulation\n",
    "\n",
    "Functions\n",
    "\n",
    "`np.random.standard_normal`, `np.random.standard_t`, `np.random.RandomState`\n",
    "\n",
    "### Exercise 5\n",
    "\n",
    "Simulate 100 standard Normal random variables\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:26.957782Z",
     "iopub.status.busy": "2023-09-28T12:33:26.957782Z",
     "iopub.status.idle": "2023-09-28T12:33:27.127917Z",
     "shell.execute_reply": "2023-09-28T12:33:27.127392Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.74853182  0.00570743 -1.43625788  0.83781064  0.04405833  0.46523284\n",
      "  0.10429603 -0.58350807  1.23537454  1.2828819 ]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "x = np.random.standard_normal(100)\n",
    "print(x[:10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 6\n",
    "\n",
    "Simulate 100 random variables from a $N\\left(.08,.2^{2}\\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.127917Z",
     "iopub.status.busy": "2023-09-28T12:33:27.127917Z",
     "iopub.status.idle": "2023-09-28T12:33:27.135532Z",
     "shell.execute_reply": "2023-09-28T12:33:27.135532Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.35717346 -0.08418996 -0.29114298 -0.01254201 -0.15909042 -0.14751249\n",
      " -0.4147758   0.13698652 -0.05896361 -0.22702702]\n"
     ]
    }
   ],
   "source": [
    "x = np.random.normal(0.08, 0.20, 100)\n",
    "print(x[:10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 7\n",
    "\n",
    "Simulate 100 random variables from a Students t with 8 degrees of freedom\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.137046Z",
     "iopub.status.busy": "2023-09-28T12:33:27.137046Z",
     "iopub.status.idle": "2023-09-28T12:33:27.143130Z",
     "shell.execute_reply": "2023-09-28T12:33:27.143130Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.7635763   4.91532809 -0.25849491  0.3148048   0.62468202 -0.25901158\n",
      " -1.65966825  1.80346246  2.00472374 -0.35256962]\n"
     ]
    }
   ],
   "source": [
    "x = np.random.standard_t(8, size=100)\n",
    "print(x[:10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 8\n",
    "\n",
    "Simulate 100 random variables from a Students t with 8 degrees of freedom with a mean of 8% and a volatility of 20%. Note: $V\\left[X\\right]=\\frac{\\nu}{\\nu-2}$ when $X\\sim t_{\\nu}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.143130Z",
     "iopub.status.busy": "2023-09-28T12:33:27.143130Z",
     "iopub.status.idle": "2023-09-28T12:33:27.151159Z",
     "shell.execute_reply": "2023-09-28T12:33:27.151159Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.05936668 -0.05321954  0.20630707  0.14163083 -0.01681002 -0.54887556\n",
      "  0.18339363 -0.27534791  0.07396293  0.00630069]\n"
     ]
    }
   ],
   "source": [
    "e = np.random.standard_t(8, size=100)\n",
    "e = e / np.sqrt(8 / (8 - 2))\n",
    "x = 0.08 + 0.2 * e\n",
    "print(x[:10])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Exercise 9\n",
    "\n",
    "Simulate two identical sets of 100 standard normal random variables by resetting the random number generator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.153170Z",
     "iopub.status.busy": "2023-09-28T12:33:27.153170Z",
     "iopub.status.idle": "2023-09-28T12:33:27.159989Z",
     "shell.execute_reply": "2023-09-28T12:33:27.159989Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0.]\n"
     ]
    }
   ],
   "source": [
    "# Legacy method for any NumPy\n",
    "rs = np.random.RandomState(30092019)\n",
    "# Modern method, NumPy >= 1.17\n",
    "rs = np.random.default_rng(30092019)\n",
    "\n",
    "x1 = rs.standard_normal(100)\n",
    "\n",
    "# rs2 = np.random.RandomState(30092019)\n",
    "rs2 = np.random.default_rng(30092019)\n",
    "\n",
    "x2 = rs2.standard_normal(100)\n",
    "print(x1 - x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.159989Z",
     "iopub.status.busy": "2023-09-28T12:33:27.159989Z",
     "iopub.status.idle": "2023-09-28T12:33:27.168327Z",
     "shell.execute_reply": "2023-09-28T12:33:27.168327Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0.]\n"
     ]
    }
   ],
   "source": [
    "# When using the legacy RandomState\n",
    "rs = np.random.RandomState(30092019)\n",
    "state = rs.get_state()\n",
    "x1 = rs.standard_normal(100)\n",
    "rs.set_state(state)\n",
    "x2 = rs.standard_normal(100)\n",
    "print(x1 - x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.168327Z",
     "iopub.status.busy": "2023-09-28T12:33:27.168327Z",
     "iopub.status.idle": "2023-09-28T12:33:27.176217Z",
     "shell.execute_reply": "2023-09-28T12:33:27.176217Z"
    },
    "jupyter": {
     "outputs_hidden": false
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\n",
      " 0. 0. 0. 0.]\n"
     ]
    }
   ],
   "source": [
    "# If using the modern way\n",
    "rs = np.random.default_rng(30092019)\n",
    "state = rs.bit_generator.state\n",
    "x1 = rs.standard_normal(100)\n",
    "rs.bit_generator.state = state\n",
    "x2 = rs.standard_normal(100)\n",
    "print(x1 - x2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### Exercise 10\n",
    "\n",
    "Repeat exercise 7 using only `standard_normal`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.176217Z",
     "iopub.status.busy": "2023-09-28T12:33:27.176217Z",
     "iopub.status.idle": "2023-09-28T12:33:27.184669Z",
     "shell.execute_reply": "2023-09-28T12:33:27.184669Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.00851377 -0.12160312  0.25513281  0.00612633 -0.01818945  0.12300775\n",
      " -0.00469669  0.0382072   0.0040533  -0.02505603]\n"
     ]
    }
   ],
   "source": [
    "# Modern way to get a reproducible random generator\n",
    "rs = np.random.default_rng(30092019)\n",
    "\n",
    "df = 8\n",
    "\n",
    "num = rs.standard_normal(100)\n",
    "denom = rs.standard_normal((100, df))\n",
    "denom = np.sum(denom**2, 1)\n",
    "denom = denom / df\n",
    "\n",
    "e = num / np.sqrt(denom)\n",
    "\n",
    "e = e / np.sqrt(df / (df - 2))\n",
    "x = 0.08 + 0.2 * e\n",
    "\n",
    "print(x[:10])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2023-09-28T12:33:27.184669Z",
     "iopub.status.busy": "2023-09-28T12:33:27.184669Z",
     "iopub.status.idle": "2023-09-28T12:33:28.207143Z",
     "shell.execute_reply": "2023-09-28T12:33:28.207143Z"
    },
    "pycharm": {
     "is_executing": false,
     "name": "#%%\n"
    },
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Kurtosis: 4.505665233546498\n",
      "Theoretical Value: 4.5\n"
     ]
    }
   ],
   "source": [
    "from scipy import stats\n",
    "\n",
    "reps = 1000000\n",
    "num = rs.standard_normal(reps)\n",
    "denom = rs.standard_normal((reps, df))\n",
    "denom = np.sum(denom**2, 1)\n",
    "denom = denom / df\n",
    "\n",
    "e = num / np.sqrt(denom)\n",
    "print(f\"Kurtosis: {3 + stats.kurtosis(e)}\")\n",
    "print(f\"Theoretical Value: {3 + 6 / (df - 4)}\")"
   ]
  }
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