Lab01_Q2.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T16:59:14.383565Z",
     "start_time": "2018-09-14T16:59:14.377260Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T16:59:14.748088Z",
     "start_time": "2018-09-14T16:59:14.741858Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "151.0 is the scattering angle (degrees)\n"
     ]
    }
   ],
   "source": [
    "# Problem 2 Part A\n",
    "f_height =  0.25 #float(input('What is the height of the particle?:'))\n",
    "# commented out inputs so that we can convert to pdf\n",
    "f_scattering_angle = np.pi - 2* np.arcsin(f_height)  # Formula from Assignment Paper\n",
    "print(np.round(f_scattering_angle*(180/np.pi),0), 'is the scattering angle (degrees)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2 Part b Pseudo Code for  Simulation of Scattering\n",
    "1. Import required packages for handling arrays and plotting tools.\n",
    "2. Generate an array of random heights ranging from $ [0, 1] $ and input them into the equation derived in Part a.\n",
    "2. Take the input array and plot it as a histogram \n",
    "3. Take the output array and plot it as a histogram \n",
    "4. Given a sufficiently large input array that it emulates true randomness, calculate the probability that the scattering angle lies between the ranges:\n",
    "    * 175 and 185 degrees vs 20 and 30 degrees \n",
    "    * 170 and 190 degrees vs 90 and 110 degrees\n",
    "5. This calculation is done by comparing the ratios of particles within each range to the the total number of particles and to each range.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T16:59:15.799312Z",
     "start_time": "2018-09-14T16:59:15.791384Z"
    }
   },
   "outputs": [],
   "source": [
    "#import required packages\n",
    "import matplotlib.pyplot as plt #import stuff for plotting\n",
    "import time \n",
    "import seaborn as sns\n",
    "sns.set()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T16:59:16.491195Z",
     "start_time": "2018-09-14T16:59:16.121799Z"
    }
   },
   "outputs": [],
   "source": [
    "i_number = 10000000 #int(input('How many particles are you shooting? :'))  # try 10000000. Not perfect but not a huge computation time\n",
    "a_height = np.random.uniform(-1, 1, i_number)            # now an array instead of single float\n",
    "a_scattering_angle = np.pi - 2* np.arcsin(a_height)  # Formula from Assignment Paper\n",
    "a_scattering_angle_deg = np.round(a_scattering_angle*(180/np.pi))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T16:59:17.006804Z",
     "start_time": "2018-09-14T16:59:16.531790Z"
    }
   },
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 504x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting Random Height Distribuition of Particles\n",
    "plt.figure(figsize = (7,4))\n",
    "plt.hist(a_height,edgecolor='black', linewidth=1.2, alpha = 0.7, color = 'mediumslateblue')\n",
    "plt.title('Random Height Distribuition of Particles')\n",
    "plt.xlabel('Height')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T17:00:40.266692Z",
     "start_time": "2018-09-14T17:00:37.943267Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plotting Angle Distribuition of Particles\n",
    "plt.figure(figsize = (10,4))\n",
    "\n",
    "data = a_scattering_angle_deg\n",
    "binwidth= 10\n",
    "bins=np.arange(min(data), max(data)+ binwidth, binwidth)\n",
    "plt.hist(data,bins  \\\n",
    "         , alpha = 0.7, color = 'salmon')\n",
    "plt.title('Scattering Angle Distribuition of Particles')\n",
    "plt.xlabel('Scattering Angle (deg)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2 Part B\n",
    "The distribuition of the angle is not uniform, but rather is something like a wide gaussian. This makes intuitive sense because the surface area near the center is smaller for corresponding height, so their are more angles within within narrow ranges at the center then near the edges. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2018-09-14T16:59:20.472359Z",
     "start_time": "2018-09-14T16:59:19.614856Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7626952743996898\n",
      "0.10628182855097044\n"
     ]
    }
   ],
   "source": [
    "# probabilities \n",
    "a_histval,a_binedge = np.histogram(data, bins) # The histogram values and bin edges for probability\n",
    "# finding the cuts I want\n",
    "i_ninety_index = np.where(np.logical_or(a_binedge == 90,a_binedge == 100)) \n",
    "i_oneseventy_index =np.where(np.logical_or(a_binedge == 170,a_binedge == 180))\n",
    "i_twenty_index =np.where(a_binedge == 20)\n",
    "# Calculating the probabilities\n",
    "f_ninety_probability = np.sum(a_histval[i_ninety_index])/np.sum(a_histval) * 100\n",
    "f_oneseventy_probability = np.sum(a_histval[i_oneseventy_index])/np.sum(a_histval) * 100\n",
    "f_twenty_probability = np.sum(a_histval[i_twenty_index])/np.sum(a_histval) * 100\n",
    "#  little bit of a jig so I can get a cut offset by 5\n",
    "a_histval_weird, a_binedge_weird = np.histogram(data,1, range = (175,185))\n",
    "# more calculating probabilities\n",
    "f_oneseventyfive_probability = np.sum(a_histval_weird)/ np.sum(a_histval)*100\n",
    "# output of ratios of probabilities\n",
    "print(f_ninety_probability/f_oneseventy_probability )\n",
    "print(f_twenty_probability/f_oneseventy_probability )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2 Part C Probalities of Scattering Angle\n",
    "Running the above code for 10000000 particles yields a probability output of  $ 76.246 \\% $ for the scattering angle will fall between  $  90 ^{\\circ} \\lt \\theta \\lt  110 ^{\\circ}   $  as  $ 170 ^{\\circ} \\lt \\theta \\lt 190 ^{\\circ}  $. Running the same code for 20000000 (double of number before) particles the probability changes neglibly to be $ 76.185 \\% $ for this angle range. \n",
    "\n",
    "Running the above code for 10000000 particles yields a probability output of $ 10.617 \\%  $ for the scattering angle will fall between  $  20 ^{\\circ} \\lt \\theta \\lt  30 ^{\\circ}   $  as  $ 175 ^{\\circ} \\lt \\theta \\lt 185 ^{\\circ}  $. Running the same code for 20000000 (double of number before) particles the probability changes neglibly to be $ 10.643 \\% $ for this angle range.\n",
    "\n",
    "In fact, re-running the code for the same number of particles numerous times, small variances in the ouput probabilities can be observed on the same order as those observed when doubling the sampling number. From this, one can safely conclude that variance observed is merely due to variances caused by the finite random sampling, and that doubling the number of samples when already sampling a large number has virtually no effect on the probability analysis. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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