hw4_241111.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "0fe6896b-c448-47ed-a376-bd624156a557",
   "metadata": {},
   "outputs": [],
   "source": [
    "# hw4-3-b)\n",
    "# modified by Moon Chan Yeong\n",
    "import random\n",
    "import numpy\n",
    "import matplotlib.pyplot as plot\n",
    "\n",
    "mu = 0  # exact mean value\n",
    "N = 10000  # sample size\n",
    "M = 10000  #  sample times\n",
    "\n",
    "mu_N = []\n",
    "\n",
    "for m in range(M):\n",
    "    x_value = []\n",
    "    x_sum = 0\n",
    "    for n in range(N):\n",
    "        x = random.choices([-1,1], weights=[0.5, 0.5])\n",
    "        x_value.append(x[0])\n",
    "        x_sum += x[0]\n",
    "    mu_N.append(x_sum/N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0453827c-623f-426b-aa92-af0b31000bd3",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mu_star: 0.00014186000000000005\n",
      "sigma_star: 0.010017524631384741\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(close=None, block=None)>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute the mean mu_star and the standard deviation sigma_star of mu_N.\n",
    "# mean mu_star\n",
    "mu_star = numpy.mean(mu_N)\n",
    "print(f\"mu_star: {mu_star}\")\n",
    "sigma_star = numpy.std(mu_N)\n",
    "print(f\"sigma_star: {sigma_star}\")\n",
    "\n",
    "# Plot the distribution of mu_N\n",
    "plot.hist(mu_N, bins=30, edgecolor='black')\n",
    "plot.title('Fig1. The distribution of mu_N (N= 10,000, M=10,000)')\n",
    "plot.xlabel('mu_N')\n",
    "plot.ylabel('Frequency')\n",
    "plot.show"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dd790845-e415-4a89-9020-49b4db5be43d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "3535876e-28be-486a-aeeb-55f6d3ba00e5",
   "metadata": {},
   "outputs": [],
   "source": [
    "# hw4-3)-c)\n",
    "import random\n",
    "import numpy\n",
    "import math \n",
    "\n",
    "mu = 0  # exact mean value\n",
    "N = 100000000  # sample size\n",
    "# M = 10000  #  sets\n",
    "\n",
    "mean_of_mu_N = []\n",
    "\n",
    "for t in range(30):\n",
    "    x_value = []\n",
    "    for n in range(N):\n",
    "        x = random.choices([-1,1], weights=[0.5, 0.5])\n",
    "        x_value.append(x[0])\n",
    "    mean_of_mu_N.append(numpy.mean(x_value))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "5438f0f3-9276-41ea-b6f4-a1e204aae3c6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-7.437333333333336e-06\n",
      "0.00010318522177564426\n"
     ]
    }
   ],
   "source": [
    "print(numpy.mean(mean_of_mu_N))\n",
    "print(numpy.std(mean_of_mu_N))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3e1169ae-2c27-48fd-8a26-2bd52a84e870",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[7.592e-05,\n",
       " 7.856e-05,\n",
       " -0.00015246,\n",
       " -6.76e-06,\n",
       " 5.764e-05,\n",
       " -0.00011956,\n",
       " -0.0001225,\n",
       " 0.00014616,\n",
       " 9.28e-05,\n",
       " 5.214e-05,\n",
       " -0.00013122,\n",
       " -0.00014648,\n",
       " 9.562e-05,\n",
       " 6.03e-05,\n",
       " 0.00023142,\n",
       " 6.66e-05,\n",
       " 0.00015124,\n",
       " -3.846e-05,\n",
       " -1.704e-05,\n",
       " 7.118e-05,\n",
       " -3.768e-05,\n",
       " 4.008e-05,\n",
       " -3.802e-05,\n",
       " 1.288e-05,\n",
       " -0.0001321,\n",
       " -0.00012526,\n",
       " -0.0001269,\n",
       " -4.124e-05,\n",
       " -8.17e-05,\n",
       " -0.00013828]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mean_of_mu_N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "9e13c1c5-9036-4713-a859-0c0b179e0f5a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# hw4-3)-c)\n",
    "# Method 2\n",
    "\n",
    "N = 10000000\n",
    "n = 10000\n",
    "m = int(N/n)  # 1000\n",
    "\n",
    "m2_list = []\n",
    "\n",
    "for t in range(30):\n",
    "    mu_N = []\n",
    "    for i in range(m):\n",
    "        x_value = []\n",
    "        for i in range(n):\n",
    "            x = random.choices([-1,1], weights=[0.5, 0.5])\n",
    "            x_value.append(x[0])\n",
    "        mu_N.append(numpy.mean(x_value))\n",
    "    m2_list.append(numpy.mean(mu_N)) # 표본 평균의 평균"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a5661d57-be91-4e8e-a363-08380cc1ed1a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.9213333333333327e-05\n",
      "0.00023405236413152412\n"
     ]
    }
   ],
   "source": [
    "print(numpy.mean(m2_list))\n",
    "print(numpy.std(m2_list))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "3ffcbc4e-5d32-4d0b-8ffd-f09446cf8c98",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0003937999999999999,\n",
       " -5.000000000000005e-06,\n",
       " 1.32000000000001e-05,\n",
       " -0.00023719999999999994,\n",
       " -0.00028720000000000004,\n",
       " 7.760000000000006e-05,\n",
       " -3.059999999999997e-05,\n",
       " -0.000152,\n",
       " 0.0003166,\n",
       " 0.00014339999999999996,\n",
       " 0.00011459999999999995,\n",
       " -0.00039879999999999994,\n",
       " -9.599999999999942e-06,\n",
       " 0.00011000000000000003,\n",
       " -0.00015880000000000007,\n",
       " -0.00032799999999999995,\n",
       " -0.00022020000000000012,\n",
       " 0.00040380000000000006,\n",
       " -8.359999999999999e-05,\n",
       " 0.0002492,\n",
       " -7.39999999999999e-06,\n",
       " 0.00034339999999999994,\n",
       " 0.000416,\n",
       " -8.480000000000001e-05,\n",
       " 8.939999999999998e-05,\n",
       " 0.00044199999999999996,\n",
       " 8.779999999999988e-05,\n",
       " -0.00029200000000000005,\n",
       " -0.00013979999999999998,\n",
       " 0.00011059999999999992]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m2_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5a485702-e7fc-4d53-877a-1906499ef85d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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