Homework - Biased Random Walk.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "import matplotlib\n",
    "matplotlib.rcParams['font.size'] = 16\n",
    "import numpy\n",
    "from math import sqrt\n",
    "import math\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far we have looked at _unbiased_ random walks, in which we have equal probability to move forward and backward. Sometimes instead we have a _biased_ random walk, where we are more likely to move in one direction than another.\n",
    "\n",
    "Let's consider a one-dimensional random walk, in which at each step there is a probability $p$ of moving in the positive direction (and therefore a probability $1-p$ of moving in the negative direction)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean is 7.97396\n",
      "Standard deviation is 6.194194210581389\n"
     ]
    },
    {
     "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": [
    "# 1 dimensional biased random walk, sampled multiple times\n",
    "P = 0.6 # probability p of positive motion\n",
    "NSTEPS = 40 # number of steps in random walk\n",
    "NSAMPLES = 50000 # number of random walks to sample\n",
    "# now we don't care about the itermediate positions; we just want to store a list of final positions\n",
    "final_positions = []\n",
    "for sample in range(NSAMPLES):\n",
    "    # reset to zero\n",
    "    position = 0\n",
    "    for step in range(NSTEPS):\n",
    "        # we use += which means \"add the following to the existing variable\"\n",
    "        if random.random() < P:\n",
    "            position += 1\n",
    "        else:\n",
    "            position += -1\n",
    "    # now we should know where we wound up\n",
    "    final_positions.append(position)\n",
    "    \n",
    "print('Mean is', numpy.mean(final_positions))\n",
    "print('Standard deviation is', numpy.std(final_positions))\n",
    "    \n",
    "# let's make a histogram with 81 bins between -40.5 and 40.5 (keeps the bin edges integral)\n",
    "plt.hist(final_positions, range=(-40.5,40.5), bins=81)\n",
    "plt.xlabel('Final position')\n",
    "plt.ylabel('Number of trials')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By changing $p$ to a few values, verify that:\n",
    "* the mean position at the end of the random walks is $2N(p-1/2)$;\n",
    "* the standard deviation of the final positions is $2\\sqrt{Np(1-p)}$.\n",
    "\n",
    "Since you're sampling from the full ensemble, there's some randomness in the number you'll get from the simulations, so don't worry if it's slightly different from the formula prediction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}