5.2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 8: Random walk with drift"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use a biased coin to simulate a random walk of 30 steps on the line. If the coin falls heads (H), take one step to the right, if it lands tails (T ), take one step left. After 30 steps, note the final position. Take Pr(H) = 0.6 and Pr(T) = 0.4. (a) Plot a sample path. (b) Make a histogram for 200 such random walks. (c) Report the sample mean. (d) Report the sample variance. (What should these be exactly?)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random \n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "def random_walk(steps = 30, head_prob = 0.6, tail_prob = 0.4):\n",
    "    output = [0]\n",
    "    for step in range(steps):\n",
    "        flip = random.random()\n",
    "        if flip > 0.4:\n",
    "            output.append(output[step-1]+1)\n",
    "        else:\n",
    "            output.append(output[step-1]-1)\n",
    "    return output\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a) sample path \n",
    "result = random_walk()\n",
    "plt.plot(result)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "#b) \n",
    "lst_historgram = []\n",
    "for step in range(200):\n",
    "    walk = random_walk()\n",
    "    # to store last element \n",
    "    lst_historgram.append(walk[-1])\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 6., 10., 21., 36., 41., 38., 30., 11.,  6.,  1.]),\n",
       " array([-5. , -3.2, -1.4,  0.4,  2.2,  4. ,  5.8,  7.6,  9.4, 11.2, 13. ]),\n",
       " <a list of 10 Patch objects>)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(lst_historgram)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample mean is, 3.28\n",
      "Sample variance is, 13.601599999999998\n"
     ]
    }
   ],
   "source": [
    "# c) sample mean \n",
    "print(\"Sample mean is,\", np.mean(lst_historgram))\n",
    "# d) sample variance \n",
    "print(\"Sample variance is,\",np.var(lst_historgram))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 9: Gambler’s ruin with time limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "def gambler_ruin(upper_bound):\n",
    "    fortune = 100 \n",
    "    counter = 0 \n",
    "    while fortune > 0 and fortune <2100 and counter < upper_bound:\n",
    "        counter += 1\n",
    "        w = random.random()\n",
    "        if w >= 0.5: \n",
    "            fortune += 1\n",
    "        else:\n",
    "            fortune -=1 \n",
    "    return fortune"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "bounds = [100,1000,10000,100000]\n",
    "variances = []\n",
    "for i in bounds:\n",
    "    experiments = []\n",
    "    for j in range(20):\n",
    "        experiments.append(gambler_ruin(i))\n",
    "    variances.append(np.var(experiments))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f9ff57d18d0>]"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(bounds,variances)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As seen in this graph, the more we increase the upper bound, the more the variance increases as well as the bias because of the variabili of the results."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}