{"id":1086,"date":"2021-02-14T08:37:06","date_gmt":"2021-02-13T23:37:06","guid":{"rendered":"https:\/\/mell0w-5phere.net\/jaded5phere\/?p=1086"},"modified":"2021-02-25T19:30:52","modified_gmt":"2021-02-25T10:30:52","slug":"random-variable-independence","status":"publish","type":"post","link":"https:\/\/mell0w-5phere.net\/jaded5phere\/2021\/02\/14\/random-variable-independence\/","title":{"rendered":"\u78ba\u7387\u5909\u6570\u304c\u72ec\u7acb\u3067\u3042\u308b\u3068\u306f"},"content":{"rendered":"

\u3000\u300c\u78ba\u7387\u5909\u6570 \\(X, Y, Z\\) \u304c\u3042\u308a\u3001\u3053\u308c\u3089\u304c\u4e92\u3044\u306b\u72ec\u7acb\u3067\u3042\u308b\u3068\u3059\u308b\u3002\u300d
\n\u3000\u3053\u3046\u3044\u3063\u305f\u8a18\u8ff0\u306f\u78ba\u7387\u30fb\u7d71\u8a08\u5206\u91ce\u3067\u306f\u983b\u7e41\u306b\u898b\u304b\u3051\u308b\u3051\u308c\u3069\u3082\u3001\u72ec\u7acb\u306e\u5b9a\u7fa9\u3063\u3066\u306a\u3093\u3060\u308d\u3046\u3068\u601d\u3063\u305f\u3089\u610f\u5916\u3068\u51fa\u3066\u3053\u306a\u3044\u3002<\/p>\n

\n

\u30fb\u4e8b\u8c61\u306e\u72ec\u7acb<\/h4>\n

\u3000\u307e\u305a\u306f\u4e8b\u8c61\u304c\u72ec\u7acb\u3067\u3042\u308b\u3068\u306f\u4f55\u306a\u306e\u304b\u3092\u8003\u3048\u3066\u307f\u3088\u3046\u3002<\/p>\n

\u3000(\u5b9a\u7fa9)<\/strong>\u3000\u4e8b\u8c61 \\(A, B\\) \u304c\u72ec\u7acb \\(\\stackrel{\\mathrm{def}}{\\Leftrightarrow} P(A\\cap B)=P(A)P(B)\\)<\/p>\n

\u3000\u3053\u308c\u306f\u3001\u66f8\u304d\u63db\u3048\u308b\u3068 \\(\\frac{P(A\\cap B)}{P(B)}=P(A)\\)\u3068\u306a\u308b\u3002\u5de6\u8fba\u306f\u300c\u4e8b\u8c61\\(B\\)\u304c\u8d77\u304d\u305f\u6642\u306b\u4e8b\u8c61\\(A\\)\u304c\u8d77\u3053\u308b\u6761\u4ef6\u4ed8\u78ba\u7387\u300d\u3067\u3042\u308b\u3051\u308c\u3069\u3082\u3001\u3053\u308c\u304c\u53f3\u8fba\u306e\u300c\u4e8b\u8c61 \\(B\\) \u306e\u767a\u751f\u306e\u5982\u4f55\u306b\u95a2\u308f\u3089\u305a\u4e8b\u8c61 \\(A\\) \u304c\u8d77\u3053\u308b\u78ba\u7387\u300d\u3068\u7b49\u3057\u3044\u3001\u3064\u307e\u308a\u300c\u4e8b\u8c61 \\(B\\) \u304c\u4e8b\u8c61 \\(A\\) \u306b\u4f55\u306e\u5f71\u97ff\u3082\u53ca\u307c\u3055\u306a\u3044\u300d\u3068\u3044\u3046\u306e\u304c\u3001\u4e8b\u8c61\u304c\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u306e\u5b9a\u7fa9\u3067\u3042\u308b\u3002<\/p>\n

\u30fb\u78ba\u7387\u5909\u6570\u306e\u72ec\u7acb<\/h4>\n

\u3000\u3055\u3066\u3001\u5148\u306e\u4e8b\u8c61\u306e\u72ec\u7acb\u306e\u5b9a\u7fa9\u306b\u304a\u3044\u3066\u3001\u4e8b\u8c61 \\(A, B\\) \u3092\u3001\u300c(\u9023\u7d9a\u306a)\u78ba\u7387\u5909\u6570 \\(X, Y\\) \u304c\u3042\u308b\u5024 \\(x, y\\) \u3092\u3068\u308b\u3053\u3068\u300d\u3068\u304a\u3044\u3066\u307f\u3088\u3046\u3002\u305d\u3046\u3059\u308b\u3068\u3001<\/p>\n

\u3000\u78ba\u7387\u5909\u6570 \\(X, Y\\) \u304c\u72ec\u7acb \\(\\stackrel{\\mathrm{def}}{\\Leftrightarrow} P(X=x\\cap Y=y)=P(X=x)P(Y=y)\\)<\/p>\n

\u3000\u3068\u306a\u308b\u3002\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3092\u4f7f\u3063\u3066\u66f8\u3051\u3070\u3001
\n$$f(x,y)=f_X(x)f_Y(y)$$
\n\u3067\u3042\u308b\u3002\u305f\u3060\u3057\u3001\\(f_X(x)\\)\u306f\u78ba\u7387\u5909\u6570 \\(Y\\) \u306e\u5024\u306b\u304b\u304b\u308f\u3089\u305a\u78ba\u7387\u5909\u6570 \\(X\\) \u304c\u5024 \\(x\\) \u3092\u3068\u308b\u78ba\u7387\u3092\u4e0e\u3048\u308b\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 (\u5468\u8fba\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570) \u3067\u3042\u308b\u3002\u6570\u5f0f\u306b\u3059\u308c\u3070
\n$$f_X(x)=\\int_{-\\infty}^{\\infty} f(x,y)\\mathrm dy$$\u3068\u306a\u308b\u3002<\/p>\n

\u30fb\u554f\u984c<\/h4>\n

\u3000\u300c\u4e92\u3044\u306b\u72ec\u7acb\u306a\u78ba\u7387\u5909\u6570 \\(X, Y, Z\\) \u304c\u3042\u308b\u3068\u304d\u3001\\(X+Y\\)\u3068\\(Z\\)\u304c\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3002\u300d\u3068\u3044\u3046\u554f\u984c\u3092\u8003\u3048\u308b\u3002
\n\u3000\u78ba\u7387\u5909\u6570 \\(X, Y, Z\\)\u304c\u4e92\u3044\u306b\u72ec\u7acb\u3068\u3044\u3046\u3053\u3068\u306f\u3001
\n\u3000$$f_X(x)=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty} f(x,y,z)\\mathrm dy\\mathrm dz$$
\n(\\(f_Y, f_Z\\) \u3082\u540c\u69d8)\u3068\u3057\u3066\u3001\u540c\u6642\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570 \\(f\\) \u304c
\n$$f(x,y,z)=f_X(x)f_Y(y)f_Z(z)$$
\n\u3068\u66f8\u3051\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3042\u308b\u3002\u3053\u3053\u3067\u3001\\(S=X+Y,\\ T=X-Y\\)\u3068\u304a\u304d\u3001\u3042\u308b\u95a2\u6570 \\(g\\), \\(g_{ST}\\) \u3068\u5b9a\u6570 \\(k\\) \u306b\u95a2\u3057\u3066
\n$$\\begin{eqnarray}f(x,y,z)&=&kg(s,t,z)\\quad\\cdots(1)\\\\
\nf_X(x)f_Y(y)&=&kg_{ST}(s,t)\\quad\\cdots(2)\\end{eqnarray}$$
\n\u304c\u6210\u308a\u7acb\u3064\u3068\u3059\u308b\u3068\u3001\\(g(s,t,z)\\) \u304c\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3042\u308b\u305f\u3081\u306b\u306f\u3001
\n$$\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}g(s,t,z)\\mathrm ds\\mathrm dt\\mathrm dz =1$$
\n\u3067\u306a\u304f\u3066\u306f\u306a\u3089\u306a\u3044\u3002\u5909\u6570\u5909\u63db\u3092\u3059\u308b\u3068\u3001<\/p>\n

$$\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}g(s,t,z)\\mathrm ds\\mathrm dt\\mathrm dz = \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\frac1kf(x,y,z)|J|\\mathrm dx\\mathrm dy\\mathrm dz$$
\n\u3053\u3053\u3067\u3001\\(J\\)\u306f\u30e4\u30b3\u30d3\u30a2\u30f3\u3067\u3042\u308a\u3001
\n$$\\begin{eqnarray}\u3000J &=& \\frac{\\partial(s,t,z)}{\\partial(x,y,z)}\\\\
\n &=& \\mathrm{det} \\left[\\begin{array}{rrr}\\frac{\\partial s}{\\partial x} &\\frac{\\partial s}{\\partial y} &\\frac{\\partial s}{\\partial z} \\\\
\n\\frac{\\partial t}{\\partial x} &\\frac{\\partial t}{\\partial y} &\\frac{\\partial t}{\\partial z} &\\\\
\n\\frac{\\partial z }{\\partial x} &\\frac{\\partial z}{\\partial y} &\\frac{\\partial z }{\\partial z} \\end{array}\\right]\\\\
\n&=& \\mathrm{det} \\left[\\begin{array}{rrr}1&1&0\\\\1&-1&0\\\\0&0&1\\end{array}\\right]\\\\
\n&=& -2\\end{eqnarray}$$
\n\u3067\u3042\u308b\u304b\u3089\u3001\\(k=2\\) \u3067\u3042\u308b\u3002<\/p>\n

\\(g_{ST}\\) \u306b\u3064\u3044\u3066\u540c\u69d8\u306b\u8a08\u7b97\u3059\u308b\u3068\u3001$$\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}g_{ST}(s,t)\\mathrm ds\\mathrm dt = 1$$
\n\u3068\u306a\u308b\u3053\u3068\u304c\u78ba\u8a8d\u3067\u304d\u308b\u304b\u3089\u3001\\(g_{ST}\\)\u306f\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3042\u308b\u3002
\n\u3000\u3055\u3066\u3001(1)\u304a\u3088\u3073(2)\u5f0f\u304b\u3089\u3001\\(g(s,t,z)=g_{ST}(s,t)f_Z(z)\\)\u3067\u3042\u308b\u3053\u3068\u306f\u5bb9\u6613\u306b\u308f\u304b\u308b\u304b\u3089\u3001\u3042\u3068\u306f \\(g_{ST}, f_Z\\) \u304c\u3001\\(g\\)\u306e\u5468\u8fba\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044\u3002\u3053\u308c\u306f\u5358\u7d14\u3067
\n$$\\begin{eqnarray}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}g(s,t,z)\\mathrm ds\\mathrm dt &=& f_Z(z)\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}g_{ST}(s,t)\\mathrm ds\\mathrm dt\\\\
\n&=&f_Z(z)\\end{eqnarray}$$\u304a\u3088\u3073$$\\begin{eqnarray}\\int_{-\\infty}^{\\infty}g(s,t,z)\\mathrm dz &=& g_{ST}(s,t)\\int_{-\\infty}^{\\infty}f_Z(z)\\mathrm dz\\\\
\n&=&g_{ST}(s,t)\\end{eqnarray}$$\u304b\u3089\u660e\u3089\u304b\u3067\u3042\u308b\u3002<\/p>\n

\u3000\u3088\u3063\u3066\u3001\\(S\\)\u3068\\(Z\\)\u306f\u72ec\u7acb\u3067\u3042\u308b\u3002\u3064\u3044\u3067\u306b\\(T\\)\u3068\\(Z\\)\u3082\u72ec\u7acb\u3067\u3042\u308b\u3053\u3068\u304c\u793a\u305b\u305f\u3002<\/p>\n

\n

\u6570\u5b66\u7684\u306b\u306f\u30ac\u30d0\u30ac\u30d0\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u8aa4\u308a\u304c\u3042\u308c\u3070\u6307\u6458\u3092\u3044\u305f\u3060\u3051\u308b\u3068\u52a9\u304b\u308a\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u3000\u300c\u78ba\u7387\u5909\u6570 \\(X, Y, Z\\) \u304c\u3042\u308a\u3001\u3053\u308c\u3089\u304c\u4e92\u3044\u306b\u72ec\u7acb\u3067\u3042\u308b\u3068\u3059\u308b\u3002\u300d \u3000\u3053\u3046\u3044\u3063\u305f\u8a18\u8ff0\u306f\u78ba\u7387\u30fb\u7d71\u8a08\u5206\u91ce\u3067\u306f\u983b\u7e41\u306b\u898b\u304b\u3051\u308b\u3051\u308c\u3069\u3082\u3001\u72ec\u7acb\u306e\u5b9a\u7fa9\u3063\u3066\u306a\u3093\u3060\u308d\u3046\u3068\u601d\u3063\u305f\u3089\u610f\u5916\u3068\u51fa\u3066\u3053\u306a\u3044\u3002 \u30fb\u4e8b\u8c61\u306e\u72ec\u7acb \u3000\u307e\u305a\u306f\u4e8b<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[87],"tags":[],"_links":{"self":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1086"}],"collection":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/comments?post=1086"}],"version-history":[{"count":10,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1086\/revisions"}],"predecessor-version":[{"id":1097,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/1086\/revisions\/1097"}],"wp:attachment":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/media?parent=1086"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/categories?post=1086"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/tags?post=1086"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}