{"id":218,"date":"2023-08-04T00:57:07","date_gmt":"2023-08-04T00:57:07","guid":{"rendered":"https:\/\/statorials.org\/ko\/snedor%e1%84%8b%e1%85%b4-%e1%84%87%e1%85%ae%e1%86%ab%e1%84%91%e1%85%a9-f\/"},"modified":"2023-08-04T00:57:07","modified_gmt":"2023-08-04T00:57:07","slug":"snedor%e1%84%8b%e1%85%b4-%e1%84%87%e1%85%ae%e1%86%ab%e1%84%91%e1%85%a9-f","status":"publish","type":"post","link":"https:\/\/statorials.org\/ko\/snedor%e1%84%8b%e1%85%b4-%e1%84%87%e1%85%ae%e1%86%ab%e1%84%91%e1%85%a9-f\/","title":{"rendered":"\uc2a4\ub124\ub370\ucf54 f \uc720\ud1b5"},"content":{"rendered":"<p>\uc774 \uae30\uc0ac\uc5d0\uc11c\ub294 Snedecor F \ubc30\ud3ec\ud310\uc774 \ubb34\uc5c7\uc778\uc9c0, \uadf8\ub9ac\uace0 \uc5b4\ub5a4 \uc6a9\ub3c4\ub85c \uc0ac\uc6a9\ub418\ub294\uc9c0 \uc124\uba85\ud569\ub2c8\ub2e4. \ub610\ud55c Snedecor F \ubd84\ud3ec \uadf8\ub798\ud504\uc640 \uadf8 \ud1b5\uacc4\uc801 \uc18d\uc131\uc774 \ubb34\uc5c7\uc778\uc9c0 \ud655\uc778\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-distribucion-f-de-snedecor\"><\/span> Snedecor F \ubd84\ud3ec\ub780 \ubb34\uc5c7\uc785\ub2c8\uae4c?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>\ud53c\uc154-\uc2a4\ub124\ub370\ucf54 F \ubd84\ud3ec<\/strong> <strong>\ub610\ub294<\/strong> \uac04\ub2e8\ud788 <strong>F \ubd84\ud3ec<\/strong> \ub77c\uace0\ub3c4 \ud558\ub294 \uc2a4\ub124\ub370\ucf54 F \ubd84\ud3ec\ub294 \ud1b5\uacc4\uc801 \ucd94\ub860, \ud2b9\ud788 \ubd84\uc0b0 \ubd84\uc11d\uc5d0 \uc0ac\uc6a9\ub418\ub294 \uc5f0\uc18d \ud655\ub960 \ubd84\ud3ec\uc785\ub2c8\ub2e4.<\/p>\n<p> Snedecor F \ubd84\ud3ec\uc758 \uc18d\uc131 \uc911 \ud558\ub098\ub294 \uc790\uc720\ub3c4\ub97c \ub098\ud0c0\ub0b4\ub294 \ub450 \uac1c\uc758 \uc2e4\uc218 \ub9e4\uac1c\ubcc0\uc218 <em>m<\/em> \ubc0f <em>n<\/em> \uac12\uc73c\ub85c \uc815\uc758\ub41c\ub2e4\ub294 \uac83\uc785\ub2c8\ub2e4. \ub530\ub77c\uc11c Snedecor \ubd84\ud3ec F\uc758 \uae30\ud638\ub294 <em>F <sub>m,n<\/sub><\/em> \uc785\ub2c8\ub2e4. \uc5ec\uae30\uc11c <em>m<\/em> \uacfc <em>n<\/em> \uc740 \ubd84\ud3ec\ub97c \uc815\uc758\ud558\ub294 \ub9e4\uac1c\ubcc0\uc218\uc785\ub2c8\ub2e4.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-b6126a79c671267450b6523ca16b4a92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"F_{m,n}\\qquad m,n>0&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;18&#8243; width=&#8221;139&#8243; style=&#8221;vertical-align: -6px;&#8221;><\/p>\n<\/p>\n<p> \uc218\ud559\uc801\uc73c\ub85c Snedecor F \ubd84\ud3ec\ub294 \ud558\ub098\uc758 \uce74\uc774-\uc81c\uacf1 \ubd84\ud3ec\uc640 \ud574\ub2f9 \uc790\uc720\ub3c4 \uc0ac\uc774\uc758 \ubaab\uc744 \ub2e4\ub978 \uce74\uc774-\uc81c\uacf1 \ubd84\ud3ec\uc640 \ud574\ub2f9 \uc790\uc720\ub3c4 \uc0ac\uc774\uc758 \ubaab\uc73c\ub85c \ub098\ub208 \uac12\uacfc \uac19\uc2b5\ub2c8\ub2e4. \ub530\ub77c\uc11c Snedecor F \ubd84\ud3ec\ub97c \uc815\uc758\ud558\ub294 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-d407869e61ca4357ffbcb40df3bd83ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{c} X\\sim \\chi_m^2\\\\[2ex] Y\\sim \\chi_n^2\\end{array}\\right\\}\\color{orange}\\bm{\\longrightarrow}\\color{black}\\ F_{m,n}= \\cfrac{X\/m}{Y\/n}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"322\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>\ucc38\uace0:<\/strong> <a href=\"https:\/\/statorials.org\/ko\/\u110f\u1161\u110b\u1175-\u110c\u1166\u1100\u1169\u11b8-\u1107\u116e\u11ab\u1111\u1169\/\">\uce74\uc774\uc81c\uacf1 \ubd84\ud3ec\ub780 \ubb34\uc5c7\uc785\ub2c8\uae4c?<\/a><\/div>\n<p> Fisher-Snedecor F \ubd84\ud3ec\ub294 \uc601\uad6d \ud1b5\uacc4\ud559\uc790 Ronald Fisher\uc640 \ubbf8\uad6d \ud1b5\uacc4\ud559\uc790 George Snedecor\uc758 \uc774\ub984\uc744 \ub530\uc11c \uba85\uba85\ub418\uc5c8\uc2b5\ub2c8\ub2e4.<\/p>\n<p> \ud1b5\uacc4\uc5d0\uc11c Fisher-Snedecor F \ubd84\ud3ec\ub294 \ub2e4\uc591\ud55c \uc6a9\ub3c4\ub85c \uc0ac\uc6a9\ub429\ub2c8\ub2e4. \uc608\ub97c \ub4e4\uc5b4, Fisher-Snedecor F \ubd84\ud3ec\ub294 \ub2e4\uc591\ud55c \uc120\ud615 \ud68c\uadc0 \ubaa8\ub378\uc744 \ube44\uad50\ud558\ub294 \ub370 \uc0ac\uc6a9\ub418\uba70 \uc774 \ud655\ub960 \ubd84\ud3ec\ub294 \ubd84\uc0b0 \ubd84\uc11d(ANOVA)\uc5d0 \uc0ac\uc6a9\ub429\ub2c8\ub2e4. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"grafica-de-la-distribucion-f-de-snedecor\"><\/span> Snedor F \ubd84\ud3ec\ub3c4<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Snedecor F \ubd84\ud3ec\uc758 \uc815\uc758\ub97c \ud655\uc778\ud55c \ud6c4 \ubc00\ub3c4 \ud568\uc218 \uadf8\ub798\ud504\uc640 \ub204\uc801 \ud655\ub960 \uadf8\ub798\ud504\uac00 \uc544\ub798\uc5d0 \ud45c\uc2dc\ub429\ub2c8\ub2e4.<\/p>\n<p> \uc544\ub798 \uadf8\ub798\ud504\uc5d0\uc11c\ub294 \uc790\uc720\ub3c4\uac00 \uc11c\ub85c \ub2e4\ub978 Snedecor F \ubd84\ud3ec\uc758 \uc5ec\ub7ec \uc608\ub97c \ubcfc \uc218 \uc788\uc2b5\ub2c8\ub2e4. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/graphique-de-distribution-f-de-snedecor.png\" alt=\"\uc2a4\ub124\ub370\ucf54 F \ubd84\ud3ec \uadf8\ub798\ud504\" class=\"wp-image-4375\" width=\"700\" height=\"419\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> \ud55c\ud3b8, \uc544\ub798 \uadf8\ub798\ud504\ub97c \ubcf4\uba74 Snedecor F \ubd84\ud3ec\uc758 \ub204\uc801\ud655\ub960\ud568\uc218 \uadf8\ub798\ud504\uac00 \ud2b9\uc131\uac12\uc5d0 \ub530\ub77c \uc5b4\ub5bb\uac8c \ubcc0\ud654\ud558\ub294\uc9c0 \ud655\uc778\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/uploads\/2023\/08\/distribution-f-de-snedecor-probabilite-cumulative.png\" alt=\"Snedecor F \ubd84\ud3ec\uc758 \ub204\uc801 \ud655\ub960\" class=\"wp-image-4377\" width=\"700\" height=\"419\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"caracteristicas-de-la-distribucion-f-de-snedecor\"><\/span> Snedecor F \ubd84\ud3ec\uc758 \ud2b9\uc131<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \ub9c8\uc9c0\ub9c9\uc73c\ub85c \uc774 \uc139\uc158\uc5d0\uc11c\ub294 Snedecor F \ubd84\ud3ec\uc758 \uac00\uc7a5 \uc911\uc694\ud55c \ud2b9\uc131\uc744 \uc81c\uc2dc\ud569\ub2c8\ub2e4.<\/p>\n<ul>\n<li> Snedecor F \ubd84\ud3ec\uc758 \uc790\uc720\ub3c4 <em>m<\/em> \uacfc <em>n\uc740<\/em> \ubd84\ud3ec\uc758 \ubaa8\uc591\uc744 \uc815\uc758\ud558\ub294 \ub450 \uac1c\uc758 \ub9e4\uac1c\ubcc0\uc218\uc785\ub2c8\ub2e4. Snedecor F \ubd84\ud3ec\uc758 \uc774\ub7ec\ud55c \ud2b9\uc131 \uac12\uc740 \uc591\uc758 \uc815\uc218\uc785\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-043df156b055e7088fa83fe733825632_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}m,n \\in \\mathbb{Z}\\\\[2ex] m,n>0\\end{array}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;54&#8243; width=&#8221;68&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> Snedecor F \ubd84\ud3ec\uc758 \uc601\uc5ed\uc740 0\ubcf4\ub2e4 \ud06c\uac70\ub098 \uac19\uc740 \ubaa8\ub4e0 \uc2e4\uc218\ub85c \uad6c\uc131\ub429\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-0b8fb6676086841f8e7454004bfb39da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\\in [0,+\\infty) \" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> 2 <em>\ubcf4\ub2e4<\/em> \ud070 <em>n<\/em> \uac12\uc758 \uacbd\uc6b0 Snedecor F \ubd84\ud3ec\uc758 \ud3c9\uade0\uc740 <em>n<\/em> \uc5d0\uc11c 2\ub97c \ube80 \uac12\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-5861b8fbc6e7aef8cac0eb21cad00c9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}X\\sim F_{m,n}\\\\[2ex] E[X]=\\cfrac{n}{n-2} \\qquad \\text{para }n>2\\end{array} &#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;75&#8243; width=&#8221;225&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> \ub9e4\uac1c\ubcc0\uc218 <em>n<\/em> \uc774 2\ubcf4\ub2e4 \ud070 \uacbd\uc6b0 Snedecor \ubd84\ud3ec F\uc758 \ubd84\uc0b0\uc740 \ub2e4\uc74c \uacf5\uc2dd\uc744 \uc801\uc6a9\ud558\uc5ec \uacc4\uc0b0\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-9a284413da889eec7ac2e9a95e255169_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c}X\\sim F_{m,n}\\\\[2ex] Var(X)=\\cfrac{2n^2\\cdot (m+n-2)}{m\\cdot (n-2)^2\\cdot (n-4)} \\qquad \\text{para }n>4\\end{array} &#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;80&#8243; width=&#8221;366&#8243; style=&#8221;vertical-align: 0px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> \ub9e4\uac1c\ubcc0\uc218 <em>m\uc774<\/em> 2\ubcf4\ub2e4 \ud070 \uacbd\uc6b0 Snedecor \ubd84\ud3ec F\uc758 \ucd5c\ube48\uac12\uc740 \ub2e4\uc74c \uc2dd\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-45d29450e457b8fa07c41227c7b407e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Mo=\\cfrac{m-2}{m}\\cdot \\cfrac{n}{n+2}\\qquad \\text{para }m>2&#8243; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;40&#8243; width=&#8221;278&#8243; style=&#8221;vertical-align: -14px;&#8221;><\/p>\n<\/p>\n<ul>\n<li> Snedecor \ubd84\ud3ec F\uc758 \ubc00\ub3c4 \ud568\uc218 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-06871a7baaf462b75f545a02eac7dd11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle P[X=x]=\\frac{\\Gamma\\left(\\frac{m+n}{2}\\right)}{\\Gamma\\left(\\frac{m}{2}\\right)\\Gamma\\left(\\frac{n}{2}\\right)}\\left(\\frac{m}{n}\\right)^{\\frac{m}{2}}\\cdot\\frac{x^{\\frac{m-2}{2}}}{\\left(1+\\frac{mx}{n}\\right)^{\\frac{m+n}{2}}}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"370\" style=\"vertical-align: -26px;\"><\/p>\n<\/p>\n<ul>\n<li> \ubcc0\uc218\uac00 \uc790\uc720\ub3c4\uac00 <em>m<\/em> \ubc0f <em>n<\/em> \uc778 Snedecor F \ubd84\ud3ec\ub97c \ub530\ub974\ub294 \uacbd\uc6b0, \ud574\ub2f9 \ubcc0\uc218\uc758 \uc5ed\uc740 \uc790\uc720\ub3c4\uac00 \ub3d9\uc77c\ud558\uc9c0\ub9cc \uac12\uc758 \uc21c\uc11c\uac00 \ubcc0\uacbd\ub418\ub294 Snedecor F \ubd84\ud3ec\ub97c \ub530\ub985\ub2c8\ub2e4.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-33587ac8fb53a8c4360dfa7f54da2792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim F_{m,n} \\ \\color{orange}\\bm{\\longrightarrow}\\color{\\black} \\ X^{-1}\\sim F_{n,m}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"264\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<ul>\n<li> \ud559\uc0dd \ubd84\ud3ec\ub294 Snedecor F \ubd84\ud3ec\uc640 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\ub97c \uac16\uc2b5\ub2c8\ub2e4. <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/statorials.org\/wp-content\/ql-cache\/quicklatex.com-3efae8413f4588d5f30ae36f2b96a4d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\\sim t_n \\ \\color{orange}\\bm{\\longrightarrow}\\color{\\black} \\ X^2\\sim F_{1,n}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"228\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<div style=\"background-color:#FFFDE7; padding-top: 10px; padding-bottom: 10px; padding-right: 20px; padding-left: 30px; border: 2.5px dashed #FFB74D; border-radius:20px;\"> <span style=\"color:#ff951b\">\u27a4<\/span> <strong>\ucc38\uace0:<\/strong> <a href=\"https:\/\/statorials.org\/ko\/\u1112\u1161\u11a8\u1109\u1162\u11bc-\u1107\u116e\u11ab\u1111\u1169\/\">\uc2a4\ud29c\ub358\ud2b8 t \ubd84\ud3ec\ub294 \ubb34\uc5c7\uc785\ub2c8\uae4c?<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae30\uc0ac\uc5d0\uc11c\ub294 Snedecor F \ubc30\ud3ec\ud310\uc774 \ubb34\uc5c7\uc778\uc9c0, \uadf8\ub9ac\uace0 \uc5b4\ub5a4 \uc6a9\ub3c4\ub85c \uc0ac\uc6a9\ub418\ub294\uc9c0 \uc124\uba85\ud569\ub2c8\ub2e4. \ub610\ud55c Snedecor F \ubd84\ud3ec \uadf8\ub798\ud504\uc640 \uadf8 \ud1b5\uacc4\uc801 \uc18d\uc131\uc774 \ubb34\uc5c7\uc778\uc9c0 \ud655\uc778\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. Snedecor F \ubd84\ud3ec\ub780 \ubb34\uc5c7\uc785\ub2c8\uae4c? \ud53c\uc154-\uc2a4\ub124\ub370\ucf54 F \ubd84\ud3ec \ub610\ub294 \uac04\ub2e8\ud788 F \ubd84\ud3ec \ub77c\uace0\ub3c4 \ud558\ub294 \uc2a4\ub124\ub370\ucf54 F \ubd84\ud3ec\ub294 \ud1b5\uacc4\uc801 \ucd94\ub860, \ud2b9\ud788 \ubd84\uc0b0 \ubd84\uc11d\uc5d0 \uc0ac\uc6a9\ub418\ub294 \uc5f0\uc18d \ud655\ub960 \ubd84\ud3ec\uc785\ub2c8\ub2e4. Snedecor F \ubd84\ud3ec\uc758 \uc18d\uc131 \uc911 \ud558\ub098\ub294 \uc790\uc720\ub3c4\ub97c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21],"tags":[],"class_list":["post-218","post","type-post","status-publish","format-standard","hentry","category-21"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.5 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>\u25b7 Snedor\uc758 \ubc30\ud3ec\ud310 F<\/title>\n<meta name=\"description\" content=\"\uc5ec\uae30\uc11c\ub294 Snedecor F \ubd84\ud3ec\uac00 \ubb34\uc5c7\uc778\uc9c0, \uc6a9\ub3c4\ub294 \ubb34\uc5c7\uc778\uc9c0, \ud2b9\uc131 \ubc0f Snedecor F \ubd84\ud3ec \uadf8\ub798\ud504\ub97c \ud655\uc778\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" 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