{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ecd454b4",
   "metadata": {},
   "source": [
    "# First simulation with Python\n",
    "\n",
    "In this notebook we will try some simple simulation tasks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3b1c26a5",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e1ab9fab",
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(8366)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a774103",
   "metadata": {},
   "source": [
    "### Simmulating a discrete random variable (stochastic variable)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "46edd01b",
   "metadata": {},
   "source": [
    "Consider the stochastic variabel X with following f(x) (pdf):\n",
    "\n",
    "| X: |0|1|2|3|\n",
    "|:--:|:--:|:--:|:--:|:--:|\n",
    "| f(x): | 0.1 | 0.3 | 0.4 | 0.2 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "994c2c69",
   "metadata": {},
   "outputs": [],
   "source": [
    "# simulate ONE realization of the random variable:\n",
    "result = np.random.choice([0,1,2,3], p=[0.1,0.3,0.4,0.2], size=1)  # we do not need replace=True - it is the default\n",
    "print(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d67474a",
   "metadata": {},
   "source": [
    "Repeat the code in the cell above. \n",
    "\n",
    "Does the value change? \n",
    "\n",
    "Is the simulation behaving as you expect?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "78be264e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# we can also \"draw\" many obervation in one go:\n",
    "sample = np.random.choice([0,1,2,3], p=[0.1,0.3,0.4,0.2], size=100)\n",
    "print(sample)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4a664cba",
   "metadata": {},
   "source": [
    "Do these values look as expected?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8bf8556b",
   "metadata": {},
   "source": [
    "Lets make a histogram:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6f8c7c9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.hist(sample, bins=[-0.5,0.5, 1.5,2.5,3.5], edgecolor='black')\n",
    "plt.xticks([0,1,2,3])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37dd7337",
   "metadata": {},
   "source": [
    "Try re-making the sample a few times.\n",
    "\n",
    "Does the histogram fit with the simulated probabilities?\n",
    "\n",
    "What happens if you make a larger sample (e.g., 1000 observations)?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c06e01a5",
   "metadata": {},
   "source": [
    "### Simulate rolling a dice"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c2aefc8",
   "metadata": {},
   "source": [
    "Consider the stochastic variabel X describing rolling a (fair) dice. \n",
    "The f(x) (pdf) is:\n",
    "\n",
    "| X: |1|2|3|4|5|6|\n",
    "|:--:|:--:|:--:|:--:|:--:|:--:|:--:|\n",
    "| f(x): | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0eb5d9aa",
   "metadata": {},
   "outputs": [],
   "source": [
    "# a single roll (try repeating a few times):\n",
    "dice_1 = np.random.choice([1,2,3,4,5,6], size=1) # we do not need to specify p if all probabilities are equal\n",
    "print(dice_1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "23bba644",
   "metadata": {},
   "outputs": [],
   "source": [
    "# 10 rolls (try repeating a few times):\n",
    "dice_10 = np.random.choice([1,2,3,4,5,6], size=10) \n",
    "print(dice_10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9980b3c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "# count the number of each outcome:\n",
    "print(np.bincount(dice_10))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbe34955",
   "metadata": {},
   "source": [
    "What do these counts represent? Can you make sense of them (compare with the values in dice_10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "22ce6720",
   "metadata": {},
   "outputs": [],
   "source": [
    "# compare the counts with the histogram:\n",
    "plt.hist(dice_10, bins=[-0.5,0.5,1.5,2.5,3.5,4.5,5.5,6.5,7.5], edgecolor='black')\n",
    "plt.xticks([0,1,2,3,4,5,6,7])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "255eb3fc",
   "metadata": {},
   "source": [
    "KAHOOT (x1)"
   ]
  }
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