{"id":1126,"date":"2020-06-17T11:19:00","date_gmt":"2020-06-17T02:19:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1126"},"modified":"2020-06-13T10:28:51","modified_gmt":"2020-06-13T01:28:51","slug":"variance-nature","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/variance-nature","title":{"rendered":"\u5206\u6563\u306e\u6027\u8cea"},"content":{"rendered":"\n

\u5b66\u7fd2\u30ec\u30d9\u30eb\uff1a\u5927\u5b66\u751f\u3000\u96e3\u6613\u5ea6\uff1a\u2605\u2606\u2606\u2606\u2606<\/span><\/p>\n\n\n\n\n\n\n\n

\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u5206\u6563\u306b\u95a2\u3059\u308b\u6027\u8cea\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002\u6761\u4ef6\u4ed8\u304d\u5206\u6563<\/a>\u306a\u3069\u306b\u3082\u3053\u306e\u6027\u8cea\u306f\u7528\u3044\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001\u662f\u975e\u3068\u3082\u899a\u3048\u3066\u304a\u304d\u305f\u3044\u5185\u5bb9\u3067\u3059\u3002\u8a3c\u660e\u3082\u8f09\u305b\u3066\u3044\u308b\u306e\u3067\u3001\u8208\u5473\u306e\u3042\u308b\u65b9\u306f\u3054\u89a7\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u5206\u6563\u306e\u6027\u8cea<\/h3>\n\n\n\n
\u5206\u6563\u306e\u6027\u8cea<\/strong><\/div>\n
\n\u78ba\u7387\u5909\u6570\\(X\\)\u304a\u3088\u3073\\(Y\\)\u306b\u5bfe\u3057\u3066\u6b21\u306e\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u305f\u3060\u3057\\(a,b,c\\)\u306f\u5b9a\u6570\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n(i)\\( \\mathrm{Var}[c] = 0 \\)<\/p>\n(ii)\\( \\mathrm{Var}[X+c] = \\mathrm{Var}[X] \\)<\/p>\n(iii)\\( \\mathrm{Var}[cX] = c^{2}\\mathrm{Var}[X] \\)<\/p>\n(iv)\\( \\mathrm{Var}[aX+bY] = a^{2}\\mathrm{Var}[X]+b^{2}\\mathrm{Var}[Y]+2ab\\mathrm{Cov}(X,Y) \\)<\/p>\n<\/div>\n

 <\/p>\n\n\n\n

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\u3053\u308c\u3089\u306f\u5206\u6563\u306e\u57fa\u672c\u6027\u8cea\u306b\u306a\u308b\u306e\u3067\u3001\u7d76\u5bfe\u306b\u899a\u3048\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>

<\/div><\/div><\/div>\n\n\n\n

\u203b\u3000\u5206\u6563\u306b\u3064\u3044\u3066\u306f\uff1c\u5206\u6563\u306e\u5b9a\u7fa9\u306e\u8a18\u4e8b\uff1e<\/a>\u3092\u53c2\u7167<\/p>\n\n\n\n

\u203b\u3000\u5171\u5206\u6563\u306b\u3064\u3044\u3066\u306f\uff1c\u5171\u5206\u6563\u306e\u5b9a\u7fa9\u306e\u8a18\u4e8b\uff1e\u3092\u53c2\u7167<\/p>\n\n\n\n

\u8a3c\u660e<\/h3>\n\n\n\n

(i)\u306f\u5f53\u305f\u308a\u524d\u3067\u3059\u3002\u5b9a\u6570\u306e\u5206\u6563\uff08\u30c7\u30fc\u30bf\u306e\u6563\u3089\u3070\u308a\uff09\u306f0\u3067\u3059\u3088\u306d\uff1f
(ii)\u304b\u3089\u8a73\u3057\u304f\u898b\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u8a3c\u660e\u306b\u306f\u5206\u6563\u306e\u5b9a\u7fa9<\/a>\u304a\u3088\u3073\u671f\u5f85\u5024\u306e\u6027\u8cea<\/a>\u3092\u7528\u3044\u3066\u3044\u307e\u3059\u3002\u308f\u304b\u3089\u306a\u3044\u65b9\u306f\u3042\u3089\u304b\u3058\u3081\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\uff01<\/p>\n\n\n\n

\u3000<\/p>\n\n\n\n

(ii)\u306b\u3064\u3044\u3066\u5de6\u8fba\u3092\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002

\\begin{align}
\\mathrm{Var}[X+c] &= \\mathrm{E}[(X+c)^{2}] – \\mathrm{E}[X+c]^{2} \\\\
&= \\mathrm{E}[X^{2}+2cX+c^{2}]-(\\mathrm{E}[X]+c)^{2} \\\\
&= \\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2} \\\\
&= \\mathrm{Var}[X]
\\end{align}<\/div><\/p>\n\n\n\n

<\/p>\n\n\n\n

(iii)\u306b\u3064\u3044\u3066\u5de6\u8fba\u3092\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002

\\begin{align}
\\mathrm{Var}[cX] &= \\mathrm{E}[(cX)^{2}] – \\mathrm{E}[cX]^{2} \\\\
&= c^{2}\\mathrm{E}[X^{2}]-(c\\mathrm{E}[X])^{2} \\\\
&= c^{2}(\\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2}) \\\\
&= c^{2}\\mathrm{Var}[X]
\\end{align}<\/div><\/p>\n\n\n\n

<\/p>\n\n\n\n

(iv)\u306b\u3064\u3044\u3066\u5de6\u8fba\u3092\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002

\\begin{align}
\\mathrm{Var}[aX+bY] &= \\mathrm{E}[(aX+bY)^{2}] – \\mathrm{E}[aX+bY]^{2} \\\\
&= \\mathrm{E}[a^{2}X^{2}+b^{2}Y^{2}+2abXY]-(a\\mathrm{E}[X]+b\\mathrm{E}[X])^{2} \\\\
&= a^{2}( \\mathrm{E}[X^{2}]-\\mathrm{E}[X]^{2} )+ b^{2}( \\mathrm{E}[Y^{2}]-\\mathrm{E}[Y]^{2} )\\\\
&\\ \\ \\ \\ + 2ab(\\mathrm{E}[XY]-\\mathrm{E}[X]\\mathrm{E}[Y]) \\\\
&= a^{2}\\mathrm{Var}[X]+b^{2}\\mathrm{Var}[Y]+2ab\\mathrm{Cov}(X,Y)
\\end{align}<\/div><\/p>\n\n\n\n

<\/p>\n\n\n\n

\u5206\u6563\u306e\u95a2\u9023\u8a18\u4e8b<\/h3>\n\n\n\n

\u30fb\u5206\u6563\u306e\u5b9a\u7fa9<\/a>\uff1a\u5206\u6563\u3092\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u6761\u4ef6\u4ed8\u304d\u5206\u6563<\/a>\uff1a\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u5909\u6570\u306e\u5206\u6563\u306e\u6c42\u3081\u65b9\u3092\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u5171\u5206\u6563<\/a>\uff1a\u5171\u5206\u6563\u3092\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u5206\u6563\u306e\u6027\u8cea<\/a>\uff1a\u5206\u6563\u306e\u6301\u3064\u6027\u8cea\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

<\/p>\n\n\n\n