{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "64a4255e",
   "metadata": {},
   "source": [
    "# Simulation from the Normal distribution\n",
    "\n",
    "In this notebook we will work with Normal distribution, using the scipy.stats subpackage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "01b8f412",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "\n",
    "# For specific probability distributions (e.g., the Normal distribution) we will use a \n",
    "# new library: \"Scipy\" (actually only the subpackage \"scipy.stats\") \n",
    "import scipy.stats as stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "305cac9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.set_printoptions(legacy='1.25')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c7140198",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[5, 2]\n"
     ]
    }
   ],
   "source": [
    "# \"Draw\" random variates (realizations/observations) from a Normal distribution with specified mean (mu) and standard deviation (sigma).\n",
    "\n",
    "# first specify mu and sigma:\n",
    "mu = 5\n",
    "sigma = 2\n",
    "print([mu, sigma])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7d5e18b",
   "metadata": {},
   "source": [
    "We can get various information about the Normal distribution from stats.norm\n",
    "\n",
    "To simulate \"random variates\" we use the .rvs method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a0047c65",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[4.86256153]\n"
     ]
    }
   ],
   "source": [
    "result = stats.norm.rvs(loc=mu, scale=sigma, size=1)\n",
    "print(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9eac660",
   "metadata": {},
   "source": [
    "Try repeating the cell above a few times and get new realizations each time. \n",
    "\n",
    "Do the results fit with your expactations (knowing mu and sigma)?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "b1ad640d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 3.90554059  3.02100882  8.17199193  2.44856281  3.83054779  4.92373744\n",
      "  3.21915803  1.69445406  4.90201233  4.82402763  6.09061119  3.16688412\n",
      "  7.01620073  3.47167133  5.21786475  5.33844278  6.41875252  6.1013975\n",
      "  2.90658294 10.49404593  5.44232709  4.45565334  4.3913228   6.77364433\n",
      "  3.86763223  4.03588992  8.18835745  2.95314015  5.54343257 -0.55372798\n",
      "  3.92724241  2.87019383  4.31242697  5.04166579  7.65427467  9.24644403\n",
      "  2.58655849  2.20563761  4.19356565  5.4565259   6.51819379  3.92164799\n",
      "  5.55815541  3.39257409  8.09478371  6.81486545  5.93224267  6.94058171\n",
      "  7.20587687  6.34987703  2.93968448  4.95042902  6.07108386  4.99614137\n",
      "  7.16427157  7.04446678 -0.43917552  8.89688714  6.10202293  3.23008836\n",
      "  9.29506062  5.64195726  3.66934356  4.63324189  3.20879162  4.78028961\n",
      "  4.27845571  2.33520152  5.28194059  5.8102824   2.90739226  6.81187102\n",
      "  5.68219726  6.93633672  4.51672842  2.01438121  3.10865065  5.87137065\n",
      "  7.14211499  7.77140528  4.43728749  3.54992461  6.84327924  5.55883343\n",
      "  3.49043077  2.6483823   1.97461037  3.81892849  6.03636719  5.88633112\n",
      "  4.04810298  5.66144815  3.47071795  3.79064462  6.81561825  4.06498625\n",
      "  3.47708948  6.74437694  4.88653753  6.48916244]\n"
     ]
    }
   ],
   "source": [
    "# Lets try to \"draw\" 100 realizations in one go:\n",
    "sample = stats.norm.rvs(loc=mu, scale=sigma, size=100)\n",
    "print(sample)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "eb29ec9a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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0depUdenSRT6fT5L03HPPafDgwcrOzlYwGNTixYtVVVWlJUuWxOI1AgCABBd1sOzdu1cjR44M7xcVFUmSpk2bprlz52rLli2SpFtuuSVi3fbt2zVixAhJUk1NjTp0+PrizmeffaYf//jH8vv9crlc6t+/vyorK3XrrbdGOx4AAGiH2nTTrUku9qYdINa46RbtCTfd4lK72M9v/pYQAAAwHsECAACMR7AAAADjESwAAMB4BAsAADAewQIAAIxHsAAAAOMRLAAAwHgECwAAMB7BAgAAjEewAAAA4xEsAADAeAQLAAAwHsECAACMR7AAAADjESwAAMB4BAsAADAewQIAAIxHsAAAAOMRLAAAwHgECwAAMB7BAgAAjEewAAAA4xEsAADAeAQLAAAwHsECAACMR7AAAADjESwAAMB4BAsAADAewQIAAIwXdbBUVlZq/Pjx8nq9cjgc2rRpU8TjlmVp7ty58nq9Sk5O1ogRI/Thhx9e8HlLS0vVu3dvOZ1O9e7dWxs3box2NAAA0E5FHSynTp1Sv379VFJS0uTjzz//vBYuXKiSkhLt2bNHbrdbo0eP1okTJ5p9zt27d2vy5MmaMmWK3n//fU2ZMkWTJk3SO++8E+14AACgHXJYlmW1erHDoY0bN2rChAmSvrq64vV6VVhYqNmzZ0uSQqGQMjIytGDBAv3kJz9p8nkmT56sYDCoN954I3zsjjvu0De+8Q2tW7fuomYJBoNyuVwKBAJKTU1t7UsCotZjTpndIwAxc2T+OLtHwGXmYj+/Y3oPS3V1tfx+v/Ly8sLHnE6nhg8frl27djW7bvfu3RFrJGnMmDEtrgEAAJePTrF8Mr/fL0nKyMiIOJ6RkaGjR4+2uK6pNeefrymhUEihUCi8HwwGWzMyAABIAHH5LSGHwxGxb1lWo2NtXePz+eRyucJbZmZm6wcGAABGi2mwuN1uSWp0ZaS+vr7RFZT/XRftmuLiYgUCgfBWW1vbhskBAIDJYhosWVlZcrvdKi8vDx9raGhQRUWFcnNzm103ZMiQiDWStHXr1hbXOJ1OpaamRmwAAKB9ivoelpMnT+rw4cPh/erqalVVVSktLU3dunVTYWGh5s2bp+zsbGVnZ2vevHm66qqrdP/994fXTJ06VV26dJHP55MkzZw5U8OGDdOCBQt01113afPmzdq2bZt27twZg5cIAAASXdTBsnfvXo0cOTK8X1RUJEmaNm2aVq9erSeeeEJffPGFHn74YX366acaNGiQtm7dqpSUlPCampoadejw9cWd3NxcrV+/Xk8//bSeeeYZXX/99dqwYYMGDRrUltcGAADaiTZ9D4tJ+B4W2IXvYUF7wvew4FKz5XtYAAAA4oFgAQAAxiNYAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABiPYAEAAMYjWAAAgPEIFgAAYDyCBQAAGI9gAQAAxiNYAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABiPYAEAAMYjWAAAgPEIFgAAYDyCBQAAGI9gAQAAxiNYAACA8TrZPQBwXo85ZXaPAAAwFFdYAACA8QgWAABgPIIFAAAYL+bB0qNHDzkcjkZbQUFBk+fv2LGjyfP//ve/x3o0AACQoGJ+0+2ePXt09uzZ8P5f//pXjR49Wvfee2+L6w4ePKjU1NTw/nXXXRfr0QAAQIKKebD8b2jMnz9f119/vYYPH97iuvT0dF1zzTWxHgcAALQDcb2HpaGhQWvWrNH06dPlcDhaPLd///7yeDwaNWqUtm/fHs+xAABAgonr97Bs2rRJn332mR588MFmz/F4PFq+fLlycnIUCoX0q1/9SqNGjdKOHTs0bNiwZteFQiGFQqHwfjAYjOXoAADAIHENlhUrVig/P19er7fZc2688UbdeOON4f0hQ4aotrZWL774YovB4vP59Nxzz8V0XgAAYKa4/Ujo6NGj2rZtmx566KGo1w4ePFiHDh1q8Zzi4mIFAoHwVltb29pRAQCA4eJ2hWXVqlVKT0/XuHHjol67f/9+eTyeFs9xOp1yOp2tHQ8AACSQuATLuXPntGrVKk2bNk2dOkX+E8XFxTp27JheffVVSdKiRYvUo0cP9enTJ3yTbmlpqUpLS+MxGgAASEBxCZZt27appqZG06dPb/RYXV2dampqwvsNDQ2aNWuWjh07puTkZPXp00dlZWUaO3ZsPEYDAAAJyGFZlmX3ELEQDAblcrkUCAQivoAOiYO/1gzY78j86H+MD7TFxX5+87eEAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABiPYAEAAMYjWAAAgPEIFgAAYDyCBQAAGI9gAQAAxiNYAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABivk90DAADM0WNOmd0jRO3I/HF2j4BLgCssAADAeAQLAAAwHsECAACMR7AAAADjESwAAMB4BAsAADAewQIAAIxHsAAAAOMRLAAAwHgECwAAMB7BAgAAjBfzYJk7d64cDkfE5na7W1xTUVGhnJwcJSUlqWfPnlq2bFmsxwIAAAksLn/8sE+fPtq2bVt4v2PHjs2eW11drbFjx+pHP/qR1qxZoz//+c96+OGHdd1112nixInxGA8AACSYuARLp06dLnhV5bxly5apW7duWrRokSTppptu0t69e/Xiiy8SLAAAQFKc7mE5dOiQvF6vsrKy9P3vf1//+Mc/mj139+7dysvLizg2ZswY7d27V6dPn252XSgUUjAYjNgAAED7FPNgGTRokF599VX98Y9/1C9+8Qv5/X7l5ubqk08+afJ8v9+vjIyMiGMZGRk6c+aMjh8/3uy/4/P55HK5wltmZmZMXwcAADBHzIMlPz9fEydO1M0336zvfOc7KisrkyS98sorza5xOBwR+5ZlNXn8vxUXFysQCIS32traGEwPAABMFJd7WP7b1VdfrZtvvlmHDh1q8nG32y2/3x9xrL6+Xp06dVLnzp2bfV6n0ymn0xnTWQEAgJni/j0soVBIBw4ckMfjafLxIUOGqLy8POLY1q1bNWDAAF1xxRXxHg8AACSAmAfLrFmzVFFRoerqar3zzju65557FAwGNW3aNElf/Shn6tSp4fNnzJiho0ePqqioSAcOHNDKlSu1YsUKzZo1K9ajAQCABBXzHwn985//1H333afjx4/ruuuu0+DBg/X222+re/fukqS6ujrV1NSEz8/KytLrr7+uxx9/XEuWLJHX69XixYv5lWYAABDmsM7f4ZrggsGgXC6XAoGAUlNT7R4HrdBjTpndIwBIQEfmj7N7BLTBxX5+87eEAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABiPYAEAAMYjWAAAgPEIFgAAYDyCBQAAGI9gAQAAxiNYAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABiPYAEAAMYjWAAAgPEIFgAAYDyCBQAAGI9gAQAAxiNYAACA8QgWAABgPIIFAAAYr5PdAwAA0BY95pTZPULUjswfZ/cICYcrLAAAwHgECwAAMB7BAgAAjBfzYPH5fBo4cKBSUlKUnp6uCRMm6ODBgy2u2bFjhxwOR6Pt73//e6zHAwAACSjmwVJRUaGCggK9/fbbKi8v15kzZ5SXl6dTp05dcO3BgwdVV1cX3rKzs2M9HgAASEAx/y2hP/zhDxH7q1atUnp6uvbt26dhw4a1uDY9PV3XXHNNrEcCAAAJLu73sAQCAUlSWlraBc/t37+/PB6PRo0ape3bt7d4bigUUjAYjNgAAED7FNdgsSxLRUVFGjp0qPr27dvseR6PR8uXL1dpaalee+013XjjjRo1apQqKyubXePz+eRyucJbZmZmPF4CAAAwgMOyLCteT15QUKCysjLt3LlTXbt2jWrt+PHj5XA4tGXLliYfD4VCCoVC4f1gMKjMzEwFAgGlpqa2aW7YIxG//AkAWoMvjvtaMBiUy+W64Od33K6wPProo9qyZYu2b98edaxI0uDBg3Xo0KFmH3c6nUpNTY3YAABA+xTzm24ty9Kjjz6qjRs3aseOHcrKymrV8+zfv18ejyfG0wEAgEQU82ApKCjQ2rVrtXnzZqWkpMjv90uSXC6XkpOTJUnFxcU6duyYXn31VUnSokWL1KNHD/Xp00cNDQ1as2aNSktLVVpaGuvxAABAAop5sCxdulSSNGLEiIjjq1at0oMPPihJqqurU01NTfixhoYGzZo1S8eOHVNycrL69OmjsrIyjR07NtbjAQCABBTXm24vpYu9aQfm4qZbAJcLbrr9mu033QIAAMQKwQIAAIwX83tY2qNE/FEFlxsBAO0JV1gAAIDxCBYAAGA8ggUAABiPYAEAAMYjWAAAgPEIFgAAYDyCBQAAGI9gAQAAxiNYAACA8QgWAABgPIIFAAAYj2ABAADGI1gAAIDxCBYAAGA8ggUAABiPYAEAAMbrZPcAAABcbnrMKbN7hKgdmT/O1n+fKywAAMB4BAsAADAewQIAAIxHsAAAAOMRLAAAwHgECwAAMB7BAgAAjEewAAAA4xEsAADAeAQLAAAwHsECAACMF7dg+fnPf66srCwlJSUpJydHb731VovnV1RUKCcnR0lJSerZs6eWLVsWr9EAAECCiUuwbNiwQYWFhXrqqae0f/9+3X777crPz1dNTU2T51dXV2vs2LG6/fbbtX//fj355JN67LHHVFpaGo/xAABAgolLsCxcuFA//OEP9dBDD+mmm27SokWLlJmZqaVLlzZ5/rJly9StWzctWrRIN910kx566CFNnz5dL774YjzGAwAACaZTrJ+woaFB+/bt05w5cyKO5+XladeuXU2u2b17t/Ly8iKOjRkzRitWrNDp06d1xRVXNFoTCoUUCoXC+4FAQJIUDAbb+hIaORf6PObPGW/xeB/iLRHfZwC4XMTrc+X881qW1eJ5MQ+W48eP6+zZs8rIyIg4npGRIb/f3+Qav9/f5PlnzpzR8ePH5fF4Gq3x+Xx67rnnGh3PzMxsw/Tth2uR3RMAANqTeH+unDhxQi6Xq9nHYx4s5zkcjoh9y7IaHbvQ+U0dP6+4uFhFRUXh/XPnzuk///mPOnfu3OK/Y5JgMKjMzEzV1tYqNTXV7nESEu9hbPA+xgbvY9vxHsZGIr2PlmXpxIkT8nq9LZ4X82C59tpr1bFjx0ZXU+rr6xtdRTnP7XY3eX6nTp3UuXPnJtc4nU45nc6IY9dcc03rB7dRamqq8f9BmY73MDZ4H2OD97HteA9jI1Hex5aurJwX85tur7zySuXk5Ki8vDzieHl5uXJzc5tcM2TIkEbnb926VQMGDGjy/hUAAHB5ictvCRUVFemXv/ylVq5cqQMHDujxxx9XTU2NZsyYIemrH+dMnTo1fP6MGTN09OhRFRUV6cCBA1q5cqVWrFihWbNmxWM8AACQYOJyD8vkyZP1ySef6Kc//anq6urUt29fvf766+revbskqa6uLuI7WbKysvT666/r8ccf15IlS+T1erV48WJNnDgxHuMZw+l06tlnn230oy1cPN7D2OB9jA3ex7bjPYyN9vg+OqwL/R4RAACAzfhbQgAAwHgECwAAMB7BAgAAjEewAAAA4xEsNvn5z3+urKwsJSUlKScnR2+99ZbdIyUUn8+ngQMHKiUlRenp6ZowYYIOHjxo91gJzefzyeFwqLCw0O5REs6xY8f0wAMPqHPnzrrqqqt0yy23aN++fXaPlVDOnDmjp59+WllZWUpOTlbPnj3105/+VOfOnbN7NKNVVlZq/Pjx8nq9cjgc2rRpU8TjlmVp7ty58nq9Sk5O1ogRI/Thhx/aM2wbESw22LBhgwoLC/XUU09p//79uv3225Wfnx/xq95oWUVFhQoKCvT222+rvLxcZ86cUV5enk6dOmX3aAlpz549Wr58ub75zW/aPUrC+fTTT3Xbbbfpiiuu0BtvvKG//e1veumllxL2m7ftsmDBAi1btkwlJSU6cOCAnn/+eb3wwgt6+eWX7R7NaKdOnVK/fv1UUlLS5OPPP/+8Fi5cqJKSEu3Zs0dut1ujR4/WiRMnLvGkMWDhkrv11lutGTNmRBzr1auXNWfOHJsmSnz19fWWJKuiosLuURLOiRMnrOzsbKu8vNwaPny4NXPmTLtHSiizZ8+2hg4davcYCW/cuHHW9OnTI47dfffd1gMPPGDTRIlHkrVx48bw/rlz5yy3223Nnz8/fOzLL7+0XC6XtWzZMhsmbBuusFxiDQ0N2rdvn/Ly8iKO5+XladeuXTZNlfgCgYAkKS0tzeZJEk9BQYHGjRun73znO3aPkpC2bNmiAQMG6N5771V6err69++vX/ziF3aPlXCGDh2qP/3pT/roo48kSe+//7527typsWPH2jxZ4qqurpbf74/4vHE6nRo+fHhCft7E7a81o2nHjx/X2bNnG/0hyIyMjEZ/ABIXx7IsFRUVaejQoerbt6/d4ySU9evX67333tOePXvsHiVh/eMf/9DSpUtVVFSkJ598Uu+++64ee+wxOZ3OiD9BgpbNnj1bgUBAvXr1UseOHXX27Fn97Gc/03333Wf3aAnr/GdKU583R48etWOkNiFYbOJwOCL2LctqdAwX55FHHtFf/vIX7dy50+5REkptba1mzpyprVu3Kikpye5xEta5c+c0YMAAzZs3T5LUv39/ffjhh1q6dCnBEoUNGzZozZo1Wrt2rfr06aOqqioVFhbK6/Vq2rRpdo+X0NrL5w3Bcolde+216tixY6OrKfX19Y0qGBf26KOPasuWLaqsrFTXrl3tHieh7Nu3T/X19crJyQkfO3v2rCorK1VSUqJQKKSOHTvaOGFi8Hg86t27d8Sxm266SaWlpTZNlJj+7//+T3PmzNH3v/99SdLNN9+so0ePyufzESyt5Ha7JX11pcXj8YSPJ+rnDfewXGJXXnmlcnJyVF5eHnG8vLxcubm5Nk2VeCzL0iOPPKLXXntNb775prKysuweKeGMGjVKH3zwgaqqqsLbgAED9IMf/EBVVVXEykW67bbbGv1K/UcffRT+Y6+4OJ9//rk6dIj8SOrYsSO/1twGWVlZcrvdEZ83DQ0NqqioSMjPG66w2KCoqEhTpkzRgAEDNGTIEC1fvlw1NTWaMWOG3aMljIKCAq1du1abN29WSkpK+IqVy+VScnKyzdMlhpSUlEb3/Fx99dXq3Lkz9wJF4fHHH1dubq7mzZunSZMm6d1339Xy5cu1fPlyu0dLKOPHj9fPfvYzdevWTX369NH+/fu1cOFCTZ8+3e7RjHby5EkdPnw4vF9dXa2qqiqlpaWpW7duKiws1Lx585Sdna3s7GzNmzdPV111le6//34bp24le39J6fK1ZMkSq3v37taVV15pfetb3+LXcaMkqclt1apVdo+W0Pi15tb53e9+Z/Xt29dyOp1Wr169rOXLl9s9UsIJBoPWzJkzrW7dullJSUlWz549raeeesoKhUJ2j2a07du3N/n/hdOmTbMs66tfbX722Wctt9ttOZ1Oa9iwYdYHH3xg79Ct5LAsy7KplQAAAC4K97AAAADjESwAAMB4BAsAADAewQIAAIxHsAAAAOMRLAAAwHgECwAAMB7BAgAAjEewAAAA4xEsAADAeAQLAAAwHsECAACM9/9CPbWGvw9l2gAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# make a histogram of the 100 observations:\n",
    "plt.hist(sample)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d8dc646",
   "metadata": {},
   "source": [
    "Does this look like a Normal distribution?\n",
    "\n",
    "Try repeating a few times. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "68e49202",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[4.968304740043604, 1.9864303468333653]\n"
     ]
    }
   ],
   "source": [
    "# calculate the sample mean and sample standard deviation:\n",
    "print([sample.mean(), sample.std(ddof=1)])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1048a5a0",
   "metadata": {},
   "source": [
    "Are these \"estimates\" close to the expected values?\n",
    "\n",
    "(they are \"estimates\" because they a calculated from the sample - i.e. from the data)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bcab5c24",
   "metadata": {},
   "source": [
    "Try increasing the sample size (for example size=1000)\n",
    "\n",
    "Draw the histogram and calculate the sample mean and sample standard deviation again. What happens when the sample size is larger?"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Pernille_Env_python_311",
   "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": "3.11.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}