InferentialStatisticsPadhAI.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "65373676-4aad-4c22-a7f4-a6e5c085b5dc",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import statistics\n",
    "import matplotlib.pyplot as plt\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfd42058-8eac-4ee1-9496-79d874be41d2",
   "metadata": {
    "tags": []
   },
   "source": [
    "## For a Discrete Random Variable_ Dice Throw"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "2e9f879d-fb19-4ecf-b5c6-7fe1da709409",
   "metadata": {},
   "outputs": [],
   "source": [
    "def generateSampleSpace(n,k):  #function that generates k element sample space for n throws of a dice\n",
    "    sampleSpace=[]\n",
    "    for i in range(k):\n",
    "        tup= list(np.random.randint(1,7,n, dtype=int))\n",
    "        sampleSpace.append(tup)\n",
    "    return sampleSpace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "63595d23-8eb1-419c-985f-a168659c4188",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.601965965965966"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_sampleSpace=generateSampleSpace(2, 1000) #generate 1000 sample space, for two consecutive die throws\n",
    "res= [np.mean(l) for l in test_sampleSpace] #calculate the mean for each sample \n",
    "statistics.variance(res) #calculate the variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "1a78f6f5-43ad-43c9-912b-9536b80f945c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5967957957957958"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_sampleSpace=generateSampleSpace(5, 1000) #generate 1000 sample space, for five consecutive die throws\n",
    "res= [np.mean(l) for l in test_sampleSpace] #calculate the mean for each sample \n",
    "statistics.variance(res) #calculate the variance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "7164ce7f-2645-4a46-8009-90b79cd06d7a",
   "metadata": {},
   "outputs": [],
   "source": [
    "varianceList=[]\n",
    "for n in range(2, 50):\n",
    "    test_sampleSpace= generateSampleSpace(n, 1000)\n",
    "    res= [np.mean(l) for l in test_sampleSpace] #calculate the mean for each sample \n",
    "    varianceList.append(statistics.variance(res))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "29935411-b60d-4908-a3f5-28b153bfc0f6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Sample Size vs Variance Plot')"
      ]
     },
     "execution_count": 30,
     "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": [
    "## PLOT\n",
    "x=[a for a in range(len(varianceList))]\n",
    "\n",
    "\n",
    "plt.Figure\n",
    "plt.plot(x, varianceList, 'bo')\n",
    "plt.title('Sample Size vs Variance Plot')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c1f312cd-8b64-4c15-a390-9b29d1388e9d",
   "metadata": {},
   "source": [
    "## For a Normal Continuous Distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "1181b44d-9af4-4e04-b66a-937e4253d96e",
   "metadata": {},
   "outputs": [],
   "source": [
    "def generateNormalSampleSpace(a,b, n=10,k=1000):  #function that generates k element sample space for n throws of a dice\n",
    "    sampleNormalSpace=[]\n",
    "    for i in range(k):\n",
    "        tupp= list(np.random.normal(loc=a, scale=b, size=n))\n",
    "        sampleNormalSpace.append(tupp)\n",
    "    return sampleNormalSpace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "7ecfb1fc-4408-4954-a69e-9392afe26514",
   "metadata": {},
   "outputs": [],
   "source": [
    "mu= [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5]\n",
    "sdFix=1\n",
    "normal_SampleSpace= generateNormalSampleSpace(2,3) #generate 1000 example sample space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "baf0ed9f-baee-497e-917f-fb3c9f66831e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8822803359913337"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res= [np.mean(l) for l in normal_SampleSpace] #calculate the mean for each sample \n",
    "statistics.variance(res)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "e3a09096-7630-46c6-96fb-05c8bd5f719e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Sample Size vs Variance Plot')"
      ]
     },
     "execution_count": 34,
     "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": [
    "## PLOT\n",
    "x=[a for a in range(len(varianceList))]\n",
    "\n",
    "\n",
    "plt.Figure\n",
    "plt.plot(x, varianceList, 'bo')\n",
    "plt.title('Sample Size vs Variance Plot')\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df535c77-b4ee-4dc1-b784-2a7af2e21017",
   "metadata": {
    "tags": []
   },
   "source": [
    "## Central Limit Theorem\n",
    "\n",
    "The sum of n random samples from a population has a Normal Distribution if the number of samples is large."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "250239d5-b2dd-45fb-9255-b8aeeb202783",
   "metadata": {},
   "outputs": [],
   "source": [
    "population= list(np.random.rand(100000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "0574a142-822d-4aab-b098-9fd27ed6ba6f",
   "metadata": {},
   "outputs": [],
   "source": [
    "def centrallimitTheorem(n, sampleSize): #n is number of samples\n",
    "    sumTemp=0\n",
    "    sampleSum=[] #Sum over n samples\n",
    "    for i in range(n):\n",
    "        sample= random.sample(population, sampleSize)\n",
    "        sumTemp= np.mean(sample)\n",
    "        sampleSum.append(sumTemp)\n",
    "        sumTemp=0\n",
    "    return sampleSum"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f255662d-f46a-4b90-b323-2098f206dfe5",
   "metadata": {},
   "source": [
    "## Histogram plot for varying number of samples."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "2c2d71a8-82cf-442d-b95a-5cbe8c2529ee",
   "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"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAACCCAYAAACpWEEoAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAAGlklEQVR4nO3dX6gc5R3G8e9TrUrphSecIKKSEyEg6VXaoNJeaK21VqFaCiVC22hTUsSWSr0J9cKiF01vFKQiBA21UNIWW2hKLSWNKaUXKTkpadNENDFVavBPNFoVQVF+XswbHDbnZOfsb3Z2J3k+sJzZd2Z33h0eZt6ZPfsbRQRmo/rYpDtg/eYAWYoDZCkOkKU4QJbiAFnK2ZPuwKnMzs7G3NzcpLthwN69e1+NiOWD7VMdoLm5Oebn5yfdDQMkPb9Quw9hluIAWYoDZClTPQY6Xcxt+uPQZZ7bfGMHPWmf90CW4gBZigNkKQ6QpThAluIAWYpP45OanKKfzrwHshQHyFIcIEtxgCzFAbIUB8hSHCBLcYAsxQGyFAfIUhwgS3GALMUBspShAZJ0iaRdkg5KOiDpB6V9maQdkg6VvzOlXZIelHRY0r8lfbr2XuvL8ockrR/fx7KuNNkDvQ/cFRGrgSuBOyStBjYBOyNiFbCzPAf4MrCqPDYCD0MVOOAe4ArgcuCeE6Gz/hoaoIh4MSL+WabfAp4CLgJuAh4riz0G3FymbwJ+EZXdwPmSLgS+BOyIiOMR8TqwA7i+zQ9j3VvSGEjSHLAG+AdwQUS8WGa9BFxQpi8C/ld72QulbbH2wXVslDQvaf7YsWNL6Z5NQOMASfok8Fvgzoh4sz4vqkqdrVTrjIgtEbE2ItYuX35SMQibMo0CJOnjVOH5ZUT8rjS/XA5NlL+vlPajwCW1l19c2hZrtx5rchYm4FHgqYi4vzZrO3DiTGo98Pta+7fK2diVwP/Loe7PwHWSZsrg+brSZj3W5J/qPwd8E9gvaV9p+xGwGfiNpA3A88DXy7wngBuAw8A7wG0AEXFc0n3AnrLcvRFxvI0PYZMzNEAR8XdAi8z+wgLLB3DHIu+1Fdi6lA7adPOVaEtxgCzFPyycEsN+oDit9YO8B7IUB8hSHCBLcYAsxQGyFAfIUhwgS3GALMUBshRfiT6FM718XRPeA1mKA2QpDpClOECW4gBZigNkKQ6QpThAluIAWYoDZCkOkKU4QJbS5LfxWyW9Iuk/tTZXJzOg2R7o55xcCMrVyQxoVqHsb8BgEQRXJzNg9DHQWKqTWf+kB9FtVicDl7jrm1EDNLbqZC5x1y+jBsjVyQxo8D/RkrYBVwOzkl6gOptydTIDmlUou2WRWa5OZr4SbTkOkKU4QJbiAFmKA2Qp/mlzTzT5mfUkCnF6D2QpDpClOECW4gBZigNkKQ6QpThAlnLGXgdy+bp2eA9kKQ6QpThAluIAWYoDZCkOkKU4QJbiAFmKA2QpDpClnLFfZZyOJvFvr94DWUrnAZJ0vaSnSxm8TcNfYdOs00OYpLOAh4AvUhWZ2iNpe0QcbHM9/qa9O13vgS4HDkfEkYh4D/gVVVk866muB9ELlbq7ouM+nNHaHmhP3VmYpI1UFV4B3pb0dAtvOwu82sL79L0P0KAf+umCzSsWauw6QENL3UXEFmBLmyuVNB8Ra9t8zz72YRz96HoMtAdYJWmlpHOAdVRl8aynOt0DRcT7kr5HVR/xLGBrRBzosg/Wrs7HQBHxBFUtxS61ekgc0TT0AdoeHlRlDc1G468yLKXXARr2tYikH0o6WO4ctFPSitq8DyTtK4/UQL5BP26VdKy2vu/U5rVyF6MGfXigtv5nJL1Rmzf6toiIXj6oBuHPApcC5wD/AlYPLPN54BNl+nbg17V5b3fYj1uBny3w2mXAkfJ3pkzPjKMPA8t/n+oEJr0t+rwHGvq1SETsioh3ytPdVNedOu/HKbR1F6Ol9uEWYNsI6zlJnwO01DsAbQD+VHt+Xrmpy25JN3fQj6+VQ+njkk5cTG3rLkaN36ccxlcCT9aaR94WU/dVxjhI+gawFriq1rwiIo5KuhR4UtL+iHh2TF34A7AtIt6V9F2qe6xdM6Z1DbMOeDwiPqi1jbwt+rwHanQHIEnXAncDX4mId0+0R8TR8vcI8Fdgzbj6ERGv1db9CPCZpXyGNvpQs46Bw1dqW0x6MJwYvJ5NNehcyUcDx08NLLOGanC5aqB9Bji3TM8ChzjFoLOFflxYm/4qsDs+GkT/t/RnpkwvG0cfynKXAc9Rrv+1sS0mHoRkiG4Anikhubu03Uu1twH4C/AysK88tpf2zwL7y4beD2wYcz9+Ahwo69sFXFZ77bep7m50GLhtXH0oz38MbB54XWpb+Eq0pfR5DGRTwAGyFAfIUhwgS3GALMUBshQHyFIcIEv5EM1N9CSv+Ql3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n=[10, 50, 100, 500, 1000, 10000]\n",
    "j=1\n",
    "for i in n:\n",
    "    sampleSumList= centrallimitTheorem(i, 10)\n",
    "    plt.subplot(2,3,j)\n",
    "    plt.hist(sampleSumList)\n",
    "    plt.show()\n",
    "    j+=1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "236150e9-fc12-4953-9b90-46f96f3f0af3",
   "metadata": {},
   "source": [
    "## Chi Square Distribution:\n",
    "Here we check out how the variance of the entire population relates to the average of the variance of a 1000 samples of size 3 each."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "db51ff12-26c9-408d-9abd-c1cd6370cbb5",
   "metadata": {},
   "outputs": [],
   "source": [
    "population= list(np.random.rand(1000))\n",
    "varPopulation= statistics.variance(population)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "aa432d23-1b3d-45bd-8b51-f0fac1b19d95",
   "metadata": {},
   "outputs": [],
   "source": [
    "varSampleCollection=[]\n",
    "varSample=0\n",
    "def chiSquare(n=1000, sampleSize=3):   #10,15 sampleSize gives a normal distribution?\n",
    "    for i in range(n):\n",
    "        sample= random.sample(population, sampleSize)\n",
    "        varSample= statistics.variance(sample)\n",
    "        varSampleCollection.append(varSample)\n",
    "        varSample=0\n",
    "    meanVariance=np.mean(varSampleCollection)\n",
    "    return (meanVariance, varSampleCollection)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "75a9d75c-dc56-4b0a-a1e8-98e27c2e9464",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.07839882343722519"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "varPopulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "769d310c-63a3-45ae-9b47-69ae4eb160a2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.07975748469984921"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sampleVariance, varSample)= chiSquare()\n",
    "sampleVariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "6c7906a6-8a10-4902-a008-48dff94bdda4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([257., 181., 154., 138.,  77.,  83.,  54.,  31.,  14.,  11.]),\n",
       " array([1.32312193e-05, 2.79994775e-02, 5.59857238e-02, 8.39719701e-02,\n",
       "        1.11958216e-01, 1.39944463e-01, 1.67930709e-01, 1.95916955e-01,\n",
       "        2.23903202e-01, 2.51889448e-01, 2.79875694e-01]),\n",
       " <BarContainer object of 10 artists>)"
      ]
     },
     "execution_count": 42,
     "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(varSample)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e0415b0-57f7-4396-b1fe-f2d5d25dc2cd",
   "metadata": {},
   "source": [
    "### Distribution of Sum of standard normal variables:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "78462192-2608-4eef-ba79-e4aae33c6a3b",
   "metadata": {},
   "outputs": [],
   "source": [
    "Z1= np.random.normal(loc=0.0, scale=1.0, size=10000)\n",
    "Z2= np.random.normal(loc=0.0, scale=1.0, size=10000)\n",
    "Z3= np.random.normal(loc=0.0, scale=1.0, size=10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "5b3333aa-b0c6-4dad-9fcb-c43a72916db2",
   "metadata": {},
   "outputs": [],
   "source": [
    "Q= Z1**2+Z2**2\n",
    "Q2= Z1**2+Z2**2+Z3**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "7efec7b8-da3e-454e-9cb0-aec775644f96",
   "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": [
    "plt.hist(Z1, 50, density=True);\n",
    "plt.title('Plotting a standard normal variable');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "4531f806-9438-46bd-8865-2dded919fe3b",
   "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": [
    "plt.hist(Z1**2, 50, density=True);\n",
    "plt.title('Plotting square of a standard normal variable');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "049b0887-7b67-49c0-8d0d-bff0d56708df",
   "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": [
    "plt.hist(Q, 50, density=True);\n",
    "plt.title('Plotting sum of squares of two standard normal variable');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df1605e2-6190-403d-ab63-1d8715b37d0e",
   "metadata": {},
   "source": [
    "**The tail is longer for Q with n=2 {Z1, Z2} as compared to Q with n=1 {Z1}**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "f8d00d01-0015-42f2-85b9-a4fe5aa6c23a",
   "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": [
    "plt.hist(Q2, 70, density=True); #Q as a sum of squares of three standard normal variables. It is a non-monotonic function.\n",
    "plt.title('Plotting sum of squares of three standard normal variable');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d4503dd-5ce4-487c-b817-8dbdffeefd24",
   "metadata": {},
   "source": [
    "The distribution shifts toward right as the number of standard normal variables (SNV) in the squared summation keeps increasing. Till it reaches the point where with enough SNV in the summation term the distribution turns normal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "595a88e8-e0c1-4bd3-8676-c2c8a761180e",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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