{"id":1087,"date":"2020-06-13T11:00:00","date_gmt":"2020-06-13T02:00:00","guid":{"rendered":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/?p=1087"},"modified":"2020-06-07T09:47:03","modified_gmt":"2020-06-07T00:47:03","slug":"mean-nature","status":"publish","type":"post","link":"https:\/\/develop-chronos.com\/statistics-top\/statistics\/mean-nature","title":{"rendered":"\u671f\u5f85\u5024\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\u671f\u5f85\u5024\u306b\u95a2\u3059\u308b\u6027\u8cea\u3092\u307e\u3068\u3081\u3066\u3044\u307e\u3059\u3002\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024<\/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

\u671f\u5f85\u5024\u306e\u6027\u8cea<\/h3>\n\n\n\n
\u671f\u5f85\u5024\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\\(c\\)\u306f\u5b9a\u6570\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n(i)\\( \\mathrm{E}[c] = c \\)<\/p>\n(ii)\\( \\mathrm{E}[X+c] = \\mathrm{E}[X]+c \\)<\/p>\n(iii)\\( \\mathrm{E}[cX] = c\\mathrm{E}[X] \\)<\/p>\n(iv)\\( \\mathrm{E}[X+Y] = \\mathrm{E}[X]+\\mathrm{E}[Y] \\)<\/p>\n<\/div>\n

 <\/p>\n\n\n\n

<\/div>

\u3053\u308c\u3089\u306f\u671f\u5f85\u5024\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

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

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

(i)\u306f\u5f53\u305f\u308a\u524d\u3067\u3059\u3002\u5b9a\u6570\u306b\u5e73\u5747\u3092\u3068\u3063\u3066\u3082\u5b9a\u6570\u306e\u307e\u307e\u3067\u3059\u3088\u306d\uff1f
(ii)\u304b\u3089\u8a73\u3057\u304f\u898b\u3066\u307f\u307e\u3057\u3087\u3046\u3002\u8a3c\u660e\u306f\u78ba\u7387\u5909\u6570\u304c\u9023\u7d9a\u578b\u3060\u3063\u305f\u5834\u5408\u3092\u60f3\u5b9a\u3057\u3066\u8a3c\u660e\u3092\u3057\u3066\u3044\u304d\u307e\u3059\u304c\u3001\u96e2\u6563\u578b\u3067\u3082\u8a3c\u660e\u306f\u540c\u3058\u3067\u3059\u3002<\/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{E}[X+c] &= \\int_{-\\infty}^{\\infty}(x+c)f(x)dx \\\\
&= \\int_{-\\infty}^{\\infty}xf(x)dx+c\\int_{-\\infty}^{\\infty}f(x)dx \\\\
&= \\mathrm{E}[X] + c
\\end{align}<\/div>
\u3053\u306e\u5909\u5f62\u306f\u671f\u5f85\u5024\u306e\u5b9a\u7fa9<\/a>\u304a\u3088\u3073\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570<\/a>\u306e\u5b9a\u7fa9\u3092\u7528\u3044\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

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

(iii)\u306b\u3064\u3044\u3066\u5de6\u8fba\u3092\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u6c42\u3081\u305f\u3044\u5f0f\u304c\u51fa\u3066\u304d\u307e\u3059\u3002

\\begin{align}
\\mathrm{E}[cX] &= \\int_{-\\infty}^{\\infty}cxf(x)dx \\\\
&= c\\int_{-\\infty}^{\\infty}xf(x)dx\\\\
&= c\\mathrm{E}[X]
\\end{align}<\/div><\/p>\n\n\n\n

\u3000<\/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{E}[X+Y] &= \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}(x+y)f_{X}(x)f_{Y}(y)dxdy \\\\
&= \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}xf_{X}(x)f_{Y}(y)dxdy+\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}yf_{X}(x)f_{Y}(y)dxdy\\\\
&= \\int_{-\\infty}^{\\infty}xf_{X}(x)dx+\\int_{-\\infty}^{\\infty}yf_{X}(y)dy \\\\
&= \\mathrm{E}[X]+\\mathrm{E}[Y]
\\end{align}<\/div>
\u305f\u3060\u3057\u3001\\(f_{X}(x)\\)\u306f\u78ba\u7387\u5909\u6570\\(X\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3042\u308a\\(f_{Y}(y)\\)\u306f\u78ba\u7387\u5909\u6570\\(Y\\)\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u25a1<\/p>\n\n\n\n

\u671f\u5f85\u5024\u306e\u95a2\u9023\u8a18\u4e8b<\/h3>\n\n\n\n

\u30fb\u671f\u5f85\u5024\u306e\u5b9a\u7fa9<\/a>\uff1a\u671f\u5f85\u5024\u3092\u5b9a\u7fa9\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u30fb\u6761\u4ef6\u4ed8\u304d\u671f\u5f85\u5024<\/a>\uff1a\u6761\u4ef6\u4ed8\u304d\u78ba\u7387\u5909\u6570\u306e\u671f\u5f85\u5024\u306e\u6c42\u3081\u65b9\u3092\u7d39\u4ecb\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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