{"id":1401,"date":"2024-07-31T09:26:09","date_gmt":"2024-07-31T09:26:09","guid":{"rendered":"http:\/\/www.statdc.com\/?p=1401"},"modified":"2025-06-11T01:23:00","modified_gmt":"2025-06-11T01:23:00","slug":"%ed%81%b0-%ec%88%98%ec%9d%98-%eb%b2%95%ec%b9%99law-of-large-numbers","status":"publish","type":"post","link":"http:\/\/www.inmining.co.kr\/?p=1401","title":{"rendered":"\ud070 \uc218\uc758 \ubc95\uce59(Law of Large Numbers)"},"content":{"rendered":"\n

\ud070 \uc218\uc758 \ubc95\uce59(Law of Large Numbers)\uc740 \ud1b5\uacc4\ud559\uacfc \ud655\ub960\ub860\uc5d0\uc11c \ub9e4\uc6b0 \uc911\uc694\ud55c \uac1c\ub150\uc73c\ub85c, \ub3d9\uc77c\ud55c \uc2e4\ud5d8\uc744 \ubc18\ubcf5\uc801\uc73c\ub85c \uc218\ud589\ud560 \ub54c, \uadf8 \ud3c9\uade0\uac12\uc774 \uc9c4\uc815\ud55c \uae30\ub313\uac12\uc5d0 \uac00\uae4c\uc6cc\uc9c4\ub2e4\ub294 \uc6d0\ub9ac\uc774\ub2e4. \uc989, \ud1b5\uacc4\uc801 \ud655\ub960\uc774 \uc218\ud559\uc801 \ud655\ub960\ub85c \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud558\uba70, \uc774 \ubc95\uce59\uc740 \ud655\ub960\ub860\uc5d0\uc11c \uc911\uc694\ud55c \ub450 \uac00\uc9c0 \ud615\ud0dc, \uc989 \uc57d\ud55c \ud070 \uc218\uc758 \ubc95\uce59\uacfc \uac15\ud55c \ud070 \uc218\uc758 \ubc95\uce59\uc73c\ub85c \ub098\ub20c \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n\n\n\n

\\(\\)<\/p>\n\n\n\n

<\/span>\uc57d\ud55c \ud070 \uc218\uc758 \ubc95\uce59 (Weak Law of Large Numbers)<\/span><\/h3>\n\n\n\n

\uc57d\ud55c \ud070 \uc218\uc758 \ubc95\uce59\uc740 \ud45c\ubcf8\uc758 \ud06c\uae30\uac00 \uc99d\uac00\ud568\uc5d0 \ub530\ub77c \ud45c\ubcf8 \ud3c9\uade0\uc774 \ud655\ub960\uc801\uc73c\ub85c \ubaa8\uc9d1\ub2e8\uc758 \uae30\ub313\uac12\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc744 \ub9d0\ud569\ub2c8\ub2e4. \uc774\ub294 \ud45c\ubcf8 \ud3c9\uade0\uc774 \uae30\ub313\uac12\uc5d0\uc11c \ud06c\uac8c \ubc97\uc5b4\ub0a0 \ud655\ub960\uc774 0\uc73c\ub85c \uc218\ub834\ud568\uc744 \uc758\ubbf8\ud569\ub2c8\ub2e4.<\/p>\n\n\n\n

\uc218\uc2dd<\/h4>\n\n\n\n

\\[ \\lim_{n \\to \\infty} P\\left( \\left| \\frac{1}{n} \\sum_{i=1}^{n} X_i – \\mu \\right| \\geq \\epsilon \\right) = 0 \\]
\uc5ec\uae30\uc11c \\(X_i\\)\ub294 \ub3c5\ub9bd\uc774\uace0 \ub3d9\uc77c\ud55c \ubd84\ud3ec\ub97c \ub530\ub974\ub294 \ud655\ub960 \ubcc0\uc218\uc774\uba70, \\(\\mu\\)\ub294 \uae30\ub313\uac12\uc785\ub2c8\ub2e4.<\/p>\n\n\n\n

<\/span>\uac15\ud55c \ud070 \uc218\uc758 \ubc95\uce59 (Strong Law of Large Numbers)<\/span><\/h3>\n\n\n\n

\uac15\ud55c \ud070 \uc218\uc758 \ubc95\uce59\uc740 \ud45c\ubcf8\uc758 \ud06c\uae30\uac00 \ubb34\ud55c\ud788 \ucee4\uc9c8 \ub54c, \ud45c\ubcf8 \ud3c9\uade0\uc774 \uac70\uc758 \ud655\uc2e4\ud558\uac8c \ubaa8\uc9d1\ub2e8\uc758 \uae30\ub313\uac12\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc744 \uc758\ubbf8\ud569\ub2c8\ub2e4. \uc774\ub294 \ud45c\ubcf8 \ud3c9\uade0\uc774 \uae30\ub313\uac12\uc5d0 \uac70\uc758 \ud655\uc2e4\ud558\uac8c \uc218\ub834\ud568\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4.<\/p>\n\n\n\n

\uc218\uc2dd<\/h4>\n\n\n\n

\\[ P\\left( \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{i=1}^{n} X_i = \\mu \\right) = 1 \\]
\uc5ec\uae30\uc11c \\(X_i\\)\ub294 \ub3c5\ub9bd\uc774\uace0 \ub3d9\uc77c\ud55c \ubd84\ud3ec\ub97c \ub530\ub974\ub294 \ud655\ub960 \ubcc0\uc218\uc774\uba70, \\(\\mu\\)\ub294 \uae30\ub313\uac12\uc785\ub2c8\ub2e4.<\/p>\n\n\n\n

<\/span>\ud070 \uc218\uc758 \ubc95\uce59\uc758 \uc2dc\uac01\uc801 \uc124\uba85<\/span><\/h3>\n\n\n\n

\ud070 \uc218\uc758 \ubc95\uce59\uc744 \uc124\uba85\ud558\ub294 \ub370\ub294 \ub3d9\uc804 \ub358\uc9c0\uae30 \uc2e4\ud5d8\uc744 \uc608\ub85c \ub4e4 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ub3d9\uc804 \ub358\uc9c0\uae30\ub97c \uc5ec\ub7ec \ubc88 \ubc18\ubcf5\ud558\uba74, \uc55e\uba74\uc774 \ub098\uc624\ub294 \ube44\uc728\uc774 \uc810\ucc28 0.5\uc5d0 \uac00\uae4c\uc6cc\uc9c0\ub294 \ud604\uc0c1\uc744 \uad00\ucc30\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n\n\n\n

\uc608\uc81c: \ub3d9\uc804 \ub358\uc9c0\uae30<\/h4>\n\n\n\n
    \n
  1. \uc0ac\uc804 \uc815\ubcf4<\/strong>: \ub3d9\uc804\uc758 \uc55e\uba74\uc774 \ub098\uc62c \ud655\ub960\uc740 0.5\uc785\ub2c8\ub2e4.<\/li>\n\n\n\n
  2. \uc2e4\ud5d8<\/strong>: \ub3d9\uc804\uc744 \uc5ec\ub7ec \ubc88 \ub358\uc9d1\ub2c8\ub2e4.<\/li>\n\n\n\n
  3. \uacb0\uacfc<\/strong>: \ub358\uc9c4 \ud69f\uc218\uac00 \ub9ce\uc544\uc9c8\uc218\ub85d, \uc55e\uba74\uc774 \ub098\uc624\ub294 \ube44\uc728\uc774 0.5\uc5d0 \uac00\uae4c\uc6cc\uc9d1\ub2c8\ub2e4.<\/li>\n<\/ol>\n\n\n\n

    <\/span>\ud070 \uc218\uc758 \ubc95\uce59\uc758 \uc911\uc694\uc131<\/span><\/h3>\n\n\n\n
      \n
    • \ud1b5\uacc4\uc801 \ucd94\uc815<\/strong>: \ud45c\ubcf8 \ud3c9\uade0\uc774 \ubaa8\uc9d1\ub2e8\uc758 \uae30\ub313\uac12\uc744 \ucd94\uc815\ud558\ub294 \ub370 \uc0ac\uc6a9\ub420 \uc218 \uc788\ub2e4\ub294 \uc810\uc5d0\uc11c \uc911\uc694\ud55c \uc6d0\ub9ac\uc785\ub2c8\ub2e4.<\/li>\n\n\n\n
    • \uc2e4\uc81c \ub370\uc774\ud130 \ubd84\uc11d<\/strong>: \ub9ce\uc740 \ub370\uc774\ud130\ub97c \uc0ac\uc6a9\ud560 \ub54c, \ud1b5\uacc4\uc801 \ucd94\uc815\uc758 \uc815\ud655\uc131\uacfc \uc2e0\ub8b0\uc131\uc744 \ubcf4\uc7a5\ud569\ub2c8\ub2e4.<\/li>\n<\/ul>\n\n\n\n

      <\/span>\uc2dc\uac01\uc801 \uc608\uc81c \ucf54\ub4dc<\/span><\/h3>\n\n\n\n

      \ub2e4\uc74c\uc740 Python\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud070 \uc218\uc758 \ubc95\uce59\uc744 \uc2dc\uac01\ud654\ud558\ub294 \uac04\ub2e8\ud55c \uc608\uc81c\uc785\ub2c8\ub2e4.<\/p>\n\n\n\n

      import numpy as np\nimport matplotlib.pyplot as plt\n\n# \ub3d9\uc804 \ub358\uc9c0\uae30 \ud568\uc218\ndef coin_toss(n):\n    return np.random.binomial(1, 0.5, n)\n\n# \ub358\uc9c4 \ud69f\uc218\nnum_trials = 10000\nresults = coin_toss(num_trials)\n\n# \ub204\uc801 \ud3c9\uade0 \uacc4\uc0b0\ncumulative_average = np.cumsum(results) \/ (np.arange(num_trials) + 1)\n\n# \uadf8\ub798\ud504 \uadf8\ub9ac\uae30\nplt.figure(figsize=(10, 6))\nplt.plot(cumulative_average, label='Cumulative Average')\nplt.axhline(y=0.5, color='r', linestyle='--', label='True Probability (0.5)')\nplt.xlabel('Number of Trials')\nplt.ylabel('Cumulative Average')\nplt.title('Law of Large Numbers')\nplt.legend()\nplt.show()<\/code><\/pre>\n\n\n\n
      \uc774 \ucf54\ub4dc\ub294 \ub3d9\uc804\uc744 10,000\ubc88 \ub358\uc84c\uc744 \ub54c \uc55e\uba74\uc774 \ub098\uc624\ub294 \ub204\uc801 \ud3c9\uade0\uc744 \uacc4\uc0b0\ud558\uace0, \uadf8 \uacb0\uacfc\ub97c \uc2dc\uac01\ud654\ud558\uc5ec \ud070 \uc218\uc758 \ubc95\uce59\uc744 \ubcf4\uc5ec\uc90d\ub2c8\ub2e4. \ub358\uc9c4 \ud69f\uc218\uac00 \ub9ce\uc544\uc9c8\uc218\ub85d \ub204\uc801 \ud3c9\uade0\uc774 0.5\uc5d0 \uac00\uae4c\uc6cc\uc9c0\ub294 \uac83\uc744 \ud655\uc778\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n","protected":false},"excerpt":{"rendered":"
      \ud070 \uc218\uc758 \ubc95\uce59(Law of Large Numbers)\uc740 \ud1b5\uacc4\ud559\uacfc \ud655\ub960\ub860\uc5d0\uc11c \ub9e4\uc6b0 \uc911\uc694\ud55c \uac1c\ub150\uc73c\ub85c, \ub3d9\uc77c\ud55c \uc2e4\ud5d8\uc744 \ubc18\ubcf5\uc801\uc73c\ub85c \uc218\ud589\ud560 \ub54c, \uadf8 \ud3c9\uade0\uac12\uc774…<\/p>\n","protected":false},"author":1,"featured_media":1402,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-1401","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-stat","wpcat-6-id"],"_links":{"self":[{"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/posts\/1401","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1401"}],"version-history":[{"count":5,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/posts\/1401\/revisions"}],"predecessor-version":[{"id":1436,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/posts\/1401\/revisions\/1436"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=\/wp\/v2\/media\/1402"}],"wp:attachment":[{"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1401"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1401"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.inmining.co.kr\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}