Lab 3.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Lab 3: Poisson Distribution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this lab, we will explore the statistics of counting.  This notebook is provided as a template, but you will need to add your own work to some cells, and certainly add many more cells."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy.misc as misc"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from matplotlib import pyplot as plt\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Part 1:  Binomial Distribution"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Taylor says that for a large number of trials (read: \"dice\") and a small chance of success (read: \"aces when the dice have very many sides\"), the binomial distribution simplifies to the Poisson distribution.\n",
      "\n",
      "Here, we will use the random number generator to simulate throwing dice.  You should change all of the parameters to see what they do."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 1000000 trials with 3 dice\n",
      "sum_losses=0.00\n",
      "sum_wins=0.00\n",
      "sum_2s=0.00\n",
      "sum_3s=0.00\n",
      "\n",
      "for i in range(1000000): \n",
      "    y=np.random.random(3)\n",
      "    yy=y[y<(1.0/6.0)]\n",
      "    if (yy.size == 1):\n",
      "        sum_wins=sum_wins+1\n",
      "    if (yy.size == 2):\n",
      "        sum_2s=sum_2s+1\n",
      "    if (yy.size == 3):\n",
      "        sum_3s=sum_3s+1\n",
      "    if (yy.size == 0):\n",
      "        sum_losses=sum_losses+1\n",
      "a=sum_wins/10000.\n",
      "b=sum_2s/10000.\n",
      "c=sum_3s/10000.\n",
      "d=sum_losses/10000."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Adjust values above and record results to reproduce Figure 10.1 in the text."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": [
        "34.8374"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "6.9065"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 6,
       "text": [
        "0.4595"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "d"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "57.7966"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "x=np.array([0,1,2,3])\n",
      "y=np.array([d,a,b,c])\n",
      "data=(x,y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.histogram(data)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "(array([5, 1, 0, 0, 0, 0, 1, 0, 0, 1]),\n",
        " array([  0.     ,   5.77966,  11.55932,  17.33898,  23.11864,  28.8983 ,\n",
        "        34.67796,  40.45762,  46.23728,  52.01694,  57.7966 ]))"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n",
      "bins=5\n",
      "winner = [0,1,2,3]\n",
      "frequencies = [d,a,b,c]\n",
      "\n",
      "pos = np.arange(len(winner))\n",
      "width = 0.1     # gives histogram aspect to the bar diagram\n",
      "\n",
      "ax = plt.axes()\n",
      "ax.set_xticks(pos + (width / 2))\n",
      "ax.set_xticklabels(winner)\n",
      "\n",
      "plt.bar(pos, frequencies, width)\n",
      "\n",
      "\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6OhQIBNTV1aUVK1aotraWf19T2NWrV5Wenq6enh6tXr1ae/fu1erVq0d9zoRdYY//RhtM\nhieeeEKzZ8+e7DGQJJ/Pp0AgIEnKyMhQXl6eLly4MMlTYTTp6emSpO7ubvX29mrOnDkJnzNhweZG\nG2BitLa26sSJEyoqKprsUTCKvr4+BQIBua6rkpIS+f3+hM+ZsGDzHlAg9bq6urRhwwbt27dPGRkZ\nkz0ORjFjxgydPHlS586d008//ZTUbeoTFuxkbrQBMH43btzQ+vXr9fLLL6uiomKyx0GS7r//fj33\n3HP69ddfE+47YcFeuXKl/vjjD7W2tqq7u1tff/211q5dO1EvD/ynxeNxbdu2TX6/X9u3b5/scZDA\nxYsXFYvFJEnXrl3ToUOHVFhYmPB5ExbsgTfa+P1+bdy4kd9gT2GbNm3S448/rpaWFi1ZskQHDhyY\n7JEwiqNHj+qrr77S4cOHVVhYqMLCQtXX10/2WBhBJBLRU089pUAgoKKiIpWXl6u0tDTh87hxBgCM\n4E+EAYARBBsAjCDYAGAEwQYAIwg2ABhBsAHACIINAEYQbAAw4v8AUANPTg161+cAAAAASUVORK5C\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f4a48cc55d0>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next, verify Taylor's claim by reproducing Figure 11.1 in the text.  Remember, let the number of dice be large and the chance of success be small, but let the average number of successes be 3."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sum_0win=0.00\n",
      "sum_1win=0.00\n",
      "sum_2win=0.00\n",
      "sum_3win=0.00\n",
      "sum_4win=0.00\n",
      "sum_5win=0.00\n",
      "sum_6win=0.00\n",
      "sum_7win=0.00\n",
      "sum_8win=0.00\n",
      "sum_9win=0.00\n",
      "\n",
      "\n",
      "for i in range(1000000): \n",
      "    y=np.random.random(20)\n",
      "    yy=y[y<(1.0/6.0)]\n",
      "    if (yy.size == 0):\n",
      "        sum_0win=sum_0win+1\n",
      "    if (yy.size == 1):\n",
      "        sum_1win=sum_1win+1\n",
      "    if (yy.size == 2):\n",
      "        sum_2win=sum_2win+1\n",
      "    if (yy.size == 3):\n",
      "        sum_3win=sum_3win+1\n",
      "    if (yy.size == 4):\n",
      "        sum_4win=sum_4win+1\n",
      "    if (yy.size == 5):\n",
      "        sum_5win=sum_5win+1\n",
      "    if (yy.size == 6):\n",
      "        sum_6win=sum_6win+1\n",
      "    if (yy.size == 7):\n",
      "        sum_7win=sum_7win+1\n",
      "    if (yy.size == 8):\n",
      "        sum_8win=sum_8win+1\n",
      "    if (yy.size == 9):\n",
      "        sum_9win=sum_9win+1\n",
      "A=sum_0win/10000.\n",
      "B=sum_1win/10000.\n",
      "C=sum_2win/10000.\n",
      "D=sum_3win/10000.\n",
      "E=sum_4win/10000.\n",
      "F=sum_5win/10000.\n",
      "G=sum_6win/10000.\n",
      "H=sum_7win/10000.\n",
      "I=sum_8win/10000.\n",
      "K=sum_9win/10000."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "2.6194"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "B"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "10.4276"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "23.7464"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "D"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": [
        "23.7464"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "E"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 16,
       "text": [
        "20.2716"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "F"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "12.9367"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "G"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 18,
       "text": [
        "6.4914"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "H"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": [
        "2.581"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "I"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "0.8363"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "K"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": [
        "0.2294"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "bins=5\n",
      "winner = [0,1,2,3,4,5,6,7,8,9]\n",
      "frequencies = [A,B,C,D,E,F,G,H,I,K]\n",
      "\n",
      "pos = np.arange(len(winner))\n",
      "width = 0.1     # gives histogram aspect to the bar diagram\n",
      "\n",
      "ax = plt.axes()\n",
      "ax.set_xticks(pos + (width / 2))\n",
      "ax.set_xticklabels(winner)\n",
      "\n",
      "plt.bar(pos, frequencies, width)\n",
      "\n",
      "\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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QIUM8zfBfqampysvLkyQlJSUpKytLx44d8zTT1VdfLUlqbm5Wa2urkpO9vxPakSNHtHXr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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f4a48325210>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Part 2:  Poisson to Gaussian"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Taylor says that the Poisson distribution for the probability of $\\nu$ counts occurring in a given time,\n",
      "$$\n",
      "P(\\nu)=e^{-\\mu} \\frac{\\mu^\\nu}{\\nu !},\n",
      "$$\n",
      "approaches the Gaussian distribution as the expected number of counts $\\mu$ increases.  Plot the Poisson distribution for various values of $\\mu$ to show that this is true.  \n",
      "\n",
      "Start by plotting the Poisson distribution with small numbers, and make notes on what you see.  Explain why the plots make sense."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "nu=np.arange(0,50,.1)\n",
      "mu=5\n",
      "p=np.exp(-mu)*(mu**nu)/misc.factorial(nu)\n",
      "mu1=10\n",
      "p1=np.exp(-mu1)*(mu1**nu)/misc.factorial(nu)\n",
      "mu2=15\n",
      "p2=np.exp(-mu2)*(mu2**nu)/misc.factorial(nu)\n",
      "mu3=30\n",
      "p3=np.exp(-mu3)*(mu3**nu)/misc.factorial(nu)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(nu,p,'r-')\n",
      "plt.plot(nu,p1,'b-')\n",
      "plt.plot(nu,p2,'g-')\n",
      "plt.plot(nu,p3,'y-')\n",
      "plt.show"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 24,
       "text": [
        "<function matplotlib.pyplot.show>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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AjQYQsf+trVPXIbUwFdtnbjd47fjxwDPPWC0Ug7ydvTFWOhb7svbROfd61NVd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\n9N6EWCYAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f4a48d9c050>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With larger values of $\\mu$ the poisson distribution approaches zero"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now overplot a Gaussian where the mean is $\\mu$ and the standard deviation is $\\sqrt{\\mu}$.  Can you verify Taylor's claim?  If so, then you have shown that the uncertainty in a count number is the square root of that number."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(nu,p,'r-')\n",
      "plt.plot(nu,p1,'b-')\n",
      "plt.plot(nu,p2,'g-')\n",
      "plt.plot(nu,p3,'y-')\n",
      "x=np.linspace(0,50,300)\n",
      "y=(1/((mu**.5)*(2*np.pi)**.5)*np.exp((-(nu-mu)**2/(2*(mu**.5)**2))))\n",
      "y1=(1/((mu1**.5)*(2*np.pi)**.5)*np.exp((-(nu-mu1)**2/(2*(mu1**.5)**2))))\n",
      "y2=(1/((mu2**.5)*(2*np.pi)**.5)*np.exp((-(nu-mu2)**2/(2*(mu2**.5)**2))))\n",
      "y3=(1/((mu3**.5)*(2*np.pi)**.5)*np.exp((-(nu-mu3)**2/(2*(mu3**.5)**2))))\n",
      "plt.plot(nu,y,color=\"black\")\n",
      "plt.plot(nu,y1,color=\"black\")\n",
      "plt.plot(nu,y2,color=\"black\")\n",
      "plt.plot(nu,y3,color=\"black\")\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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prPn+/SE8HIqK1De2K9682Ru7l8y7mcz1mcvdu3d5++232bBhA0ZGRnVqa6zr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\nzsZudYoAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f4a48abeed0>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As mu gets larger the poisson resembles the gaussian more. This is because $\\sqrt{\\mu}$ is closer to the standard deviation. Also the Poisson becomes more bell shaped and symmetric as $\\mu$ gets larger, while the Gaussian is always bell shaped and symmetric."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Part 3:  Measuring Geiger Counts"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Measure the number of decays of C60 in 2 seconds using the instrument provided.  Now repeat the experiment 100 times.  Feel free to collaborate!  You may find it easy to save the data in a file."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data=np.loadtxt(\"ThursdayNuclearData.csv\", delimiter='|')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "data"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "array([  99.,   92.,  103.,   92.,  113.,   95.,  108.,   85.,  106.,\n",
        "         97.,  109.,  107.,   95.,  101.,   97.,  103.,  119.,   97.,\n",
        "        111.,  118.,  105.,  102.,  111.,   96.,  105.,  106.,   98.,\n",
        "         97.,   90.,   98.,  107.,   99.,  101.,  109.,  107.,  114.,\n",
        "        122.,  100.,  103.,  119.,   94.,   95.,  106.,   97.,  108.,\n",
        "         93.,  106.,   85.,  104.,   81.,   96.,  132.,   89.,   96.,\n",
        "        109.,   98.,   98.,   73.,  104.,  107.,  119.,   82.,  108.,\n",
        "        110.,  111.,  119.,  120.,  107.,  119.,   92.,   97.,   98.,\n",
        "        105.,  136.,  100.,   73.,  103.,  103.,  100.,   94.,   99.,\n",
        "        120.,  115.,  103.,   89.,   88.,   89.,  111.,  111.,   93.,\n",
        "        118.,   95.,  127.,  106.,  110.,   99.,  114.,   99.,   98.,\n",
        "         80.,   92.,   96.])"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x=np.array(data)\n",
      "# x=np.loadtxt(\"nucleardata.txt\")\n",
      "x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 28,
       "text": [
        "array([  99.,   92.,  103.,   92.,  113.,   95.,  108.,   85.,  106.,\n",
        "         97.,  109.,  107.,   95.,  101.,   97.,  103.,  119.,   97.,\n",
        "        111.,  118.,  105.,  102.,  111.,   96.,  105.,  106.,   98.,\n",
        "         97.,   90.,   98.,  107.,   99.,  101.,  109.,  107.,  114.,\n",
        "        122.,  100.,  103.,  119.,   94.,   95.,  106.,   97.,  108.,\n",
        "         93.,  106.,   85.,  104.,   81.,   96.,  132.,   89.,   96.,\n",
        "        109.,   98.,   98.,   73.,  104.,  107.,  119.,   82.,  108.,\n",
        "        110.,  111.,  119.,  120.,  107.,  119.,   92.,   97.,   98.,\n",
        "        105.,  136.,  100.,   73.,  103.,  103.,  100.,   94.,   99.,\n",
        "        120.,  115.,  103.,   89.,   88.,   89.,  111.,  111.,   93.,\n",
        "        118.,   95.,  127.,  106.,  110.,   99.,  114.,   99.,   98.,\n",
        "         80.,   92.,   96.])"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Find the mean and standard deviation of the data.  Experiment with the histogram plots until you come up with a binning that you feel best represents the data."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg=x.sum()/x.size\n",
      "std=np.sqrt(np.sum((x-x.mean())**2)/x.size)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 30,
       "text": [
        "102.5"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "std"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 31,
       "text": [
        "11.30655718130491"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hist, bin_edges = np.histogram(x,density= 1)\n",
      "bin_centres = (bin_edges[:-1] + bin_edges[1:])/2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(x,bins=15,normed=True)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x7f4a484b6390>"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How does the standard deviation of the measurements compare to the square root of the mean number of counts?  Comment in markdown."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The standard deviation is 11.3, while the root of the mean is 10.12, so the two values are close (off by 1.18), but it shows that the of error in each measurement lead to error in the mean."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, how does the propagated error in each measurement contribute to the error of the mean in the averaging process?  Show your work, and comment in markdown."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We start with the standard deviation of error\n",
      "\n",
      "$\\sigma_\\bar{x}=\\frac{\\sigma_x}{\\sqrt{N}}$\n",
      "\n",
      "for this our answer is \n",
      "\n",
      "$\\sigma_\\bar{x}=$ 1.11"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The numerator is the sum of the errors for each measurement added up in quadrature.\n",
      "\n",
      "$\\sigma_\\bar{x}=\\frac{\\sqrt{(\\delta x_1)^2+...+ (\\delta x_N)^2}}{{\\sqrt{N}}}$\n",
      "\n",
      "this is equivalent to the following statements."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Let each measurement and its error be\n",
      "\n",
      "$x_1 \\pm \\delta x_1 ,..., x_N \\pm \\delta x_N$\n",
      "\n",
      "so finding the mean of $N$ measurements we have, adding the uncertainties up in quadrature\n",
      "\n",
      "$\\mu \\pm \\delta \\mu = \\frac{x_1 +...+ x_N} {N} \\pm \\sqrt{(\\delta x_1)^2+...+ (\\delta x_N)^2}$\n",
      "\n",
      "$= \\mu \\pm \\sqrt{(\\delta x_1)^2+...+ (\\delta x_N)^2}$\n",
      "\n",
      "$= \\mu \\pm \\frac{\\sqrt{ \\frac{ \\sum_i(x_i-\\bar{x})^2}{N}}}{\\sqrt{N}}$\n",
      "\n",
      "so we have,\n",
      "\n",
      "$\\sigma_\\bar{x}=\\frac{\\sqrt{ \\frac{ \\sum_i(x_i-\\bar{x})^2}{N}}}{\\sqrt{N}}$\n",
      "\n",
      "$\\sigma_\\bar{x}=\\frac{\\sigma_x}{\\sqrt{N}}$\n",
      "\n",
      "Which shows us how the the propigation of error contributes to the error of the mean.\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this case we will have 102 values so there are 102 different individual propigated errors, so we have \n",
      "\n",
      "$\\mu= 102.5 \\pm \\sqrt{ \\frac{ \\sum(x_i-102.5)^2}{102}}$\n",
      "\n",
      "In this case we see that the error in each measuremeant is one in 102 so that each measurement contributes about 1 percent of the error. We see this correlates to or calculations; the square root of our mean is 10.12 and or standard deviation is 11.3. The difference in those values is 1.18, if we divide this by our total, we get 1.15 percent, this also shows that each measurement contributes about 1 percent of the error in our mean."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Repeat the exercises above for just the first 20 points.  What do you notice?  How do the mean and standard deviation compare?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "np.mean(x[0:20])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 35,
       "text": [
        "102.34999999999999"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg1=x[0:20].sum()/x[0:20].size\n",
      "std1=np.sqrt(np.sum((x[0:20]-x[0:20].mean())**2)/x[0:20].size)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "102.34999999999999"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For twenty points the mean is only 0.15 off that of the total sample size."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "std1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "8.7879178421284756"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The standard deviation is about 3 less than that of"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.hist(x[0:20],bins=15,normed=True)\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAX4AAAEACAYAAAC08h1NAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nUe6VtZ9AgwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7f4a48ad17d0>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Imagine that each 20-run set is one experiment.  Find the mean, the standard deviation, and standard deviation of the mean for each  of the resulting five experiments.  Show the results in a table.  Then calculate the standard deviation of the resulting five values of the mean.  Which values best represent the observed spread in the average value of the 5 experiments."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg1=x[0:20].sum()/x[0:20].size\n",
      "std1=np.sqrt(np.sum((x[0:20]-x[0:20].mean())**2)/x[0:20].size)\n",
      "stdom1=std1/(np.sqrt(20))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 41,
       "text": [
        "102.34999999999999"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "std1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "8.7879178421284756"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "stdom1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": [
        "1.9650381675682536"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg2=x[20:40].sum()/x[20:40].size\n",
      "std2=np.sqrt(np.sum((x[20:40]-x[20:40].mean())**2)/x[20:40].size)\n",
      "stdom2=std2/(np.sqrt(20))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "104.45"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "std2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 46,
       "text": [
        "7.6777275283771296"
       ]
      }
     ],
     "prompt_number": 46
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "stdom2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 47,
       "text": [
        "1.7167920666172707"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg3=x[40:60].sum()/x[40:60].size\n",
      "std3=np.sqrt(np.sum((x[40:60]-x[40:60].mean())**2)/x[40:60].size)\n",
      "stdom3=std3/(np.sqrt(20))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "98.549999999999997"
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     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "std3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 50,
       "text": [
        "11.989474550621473"
       ]
      }
     ],
     "prompt_number": 50
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    {
     "cell_type": "code",
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     "input": [
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     "metadata": {},
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      {
       "metadata": {},
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       "prompt_number": 51,
       "text": [
        "2.6809280109693354"
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     "prompt_number": 51
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg4=x[60:80].sum()/x[60:80].size\n",
      "std4=np.sqrt(np.sum((x[60:80]-x[60:80].mean())**2)/x[60:80].size)\n",
      "stdom4=std4/(np.sqrt(20))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
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       "prompt_number": 53,
       "text": [
        "104.8"
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     "prompt_number": 53
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
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     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 54,
       "text": [
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     "prompt_number": 54
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    {
     "cell_type": "code",
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     "input": [
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     "metadata": {},
     "outputs": [
      {
       "metadata": {},
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       "prompt_number": 55,
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     "prompt_number": 55
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg5=x[80:100].sum()/x[80:100].size\n",
      "std5=np.sqrt(np.sum((x[80:100]-x[80:100].mean())**2)/x[80:100].size)\n",
      "stdom5=std5/(np.sqrt(20))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "avg5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 57,
       "text": [
        "103.2"
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     "prompt_number": 57
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "std5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 58,
       "text": [
        "12.089665007765928"
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     "prompt_number": 58
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    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "stdom5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$Averages$|$Standard..Deviation$|$Standard..Deviation..of..means$\n",
      "----------|---------------------|--------------------------------\n",
      "102.35    |8.79                 |1.97                        \n",
      "104.45    |7.68                 |1.72                        \n",
      "98.55     |11.99                |2.68                        \n",
      "104.8     |13.89                |3.11                        \n",
      "103.2     |12.09                |2.70                        "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "AVG=(avg1+avg2+avg3+avg4+avg5)/5\n",
      "AVG"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 66,
       "text": [
        "102.67"
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "STD=np.sqrt(((avg1-AVG)**2+(avg2-AVG)**2+(avg3-AVG)**2+(avg4-AVG)**2+(avg5-AVG)**2)/5)\n",
      "STD"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 67,
       "text": [
        "2.2388836503936522"
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The fifth value best represents the spread of the average value for the five experiments, because the mean, and standard are most close to those of the total."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What if, instead, you simply ran five 20 $\\times$ 2 second = 40 second experiments?  What would be the uncertainty in each experiment?  Now use propagation of error to report the \"2 second\" count rate for each experiment.  What do you notice?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the range of 0 to 40 (first set of two) the error is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a=((avg1*40)**.5)/20\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 61,
       "text": [
        "3.199218654609278"
       ]
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the range of 40 to 80 (second set of two) the error is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b=((avg2*40)**.5)/20\n",
      "b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 62,
       "text": [
        "3.2318725222384623"
       ]
      }
     ],
     "prompt_number": 62
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the range of 80 to 120 (third set of two) the error is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c=((avg3*40)**.5)/20\n",
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 63,
       "text": [
        "3.1392674304684527"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the range of 120 to 160 (fourth set of two) the error is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "d=((avg4*40)**.5)/20\n",
      "d"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 64,
       "text": [
        "3.2372828112477294"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the range of 160 to 200 (fifth set of two) the error is"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "e=((avg5*40)**.5)/20\n",
      "e"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 65,
       "text": [
        "3.212475680841802"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We see that when the number of seconds in each experiment is increased, the standard deviation of mean for each experiment starts to converge at a common value, around 3.\n",
      "For the most part this is not suprising, because with more trials (that have trending data), there is usually less fluctuation in the mean."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}