\n\u3000\u81ea\u5206\u7528\u5099\u5fd8\u9332<\/p>\n
\u3000<\/p>\n
$$f(x)=ae^{-b(x-c)^2} (a,b,c\\in \\mathbb R)$$\u306e\u5f62\u3092\u3057\u305f\u95a2\u6570\u3092\u30ac\u30a6\u30b9\u95a2\u6570\u3068\u3044\u3046\u3089\u3057\u3044\u3002
\n\\(x-c=\\xi\\) \u3068\u7f6e\u304d\u63db\u3048\u308c\u3070\u3001\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f\u3088\u308a
\n$$\\int^\\infty_{-\\infty} f(x)\\mathrm dx=\\int^\\infty_{-\\infty} ae^{-b\\xi^2}\\mathrm d\\xi=a\\sqrt{\\frac \\pi b }\\quad \\cdots (1)$$\u3067\u3042\u308b\u3002<\/p>\n
\u3000<\/p>\n
\u3000\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c\u4e0a\u8a18\u306e\u30ac\u30a6\u30b9\u95a2\u6570\u306b\u306a\u308b\u5206\u5e03\u304c\u30ac\u30a6\u30b9\u5206\u5e03\u3001\u3044\u308f\u3086\u308b\u6b63\u898f\u5206\u5e03\u3067\u3042\u308b\u3002
\n\u3000\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u3067\u3042\u308b\u306e\u3067\u3001\u533a\u9593\\((-\\infty,\\infty)\\)\u306b\u304a\u3051\u308b\u7a4d\u5206\u5024\u306f\\(1\\)\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308b\u3002\u3059\u306a\u308f\u3061\u3001(1)\u5f0f\u3088\u308a\u3001\\(\\displaystyle a\\sqrt{\\frac \\pi b }=1 \\quad \\cdots (2)\\)\u3067\u3042\u308b\u3002
\n\u3000
\n\u3000<\/p>\n
\u3000\u671f\u5f85\u5024\\(\\mu\\)\u3092$$\\mu=\\int^\\infty_{-\\infty} xf(x)\\mathrm dx$$\u3068\u5b9a\u7fa9\u3059\u308b\u3068\u304d\u3001\u5206\u6563\\(\\sigma^2\\)\u306f\\((x-\\mu)^2\\)\u306e\u671f\u5f85\u5024\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3059\u306a\u308f\u3061\u3001$$\\sigma^2=\\int^\\infty_{-\\infty} (x-\\mu)^2f(x)\\mathrm dx$$\u3067\u3042\u308b\u3002<\/p>\n
\u3000<\/p>\n
\u3000\u307e\u305a\u671f\u5f85\u5024\u304b\u3089\u8003\u3048\u3066\u3044\u304f\u3002\u5148\u307b\u3069\u3068\u540c\u69d8\u306b\\(x-c=\\xi\\) \u3068\u7f6e\u304d\u63db\u3048\u308c\u3070\u3001
\n$$\\begin{eqnarray*}\\mu&=&\\int^\\infty_{-\\infty} (\\xi+c)ae^{-b\\xi^2}\\mathrm d\\xi\\\\
\n&=&\\int^\\infty_{-\\infty} \\xi ae^{-b\\xi^2}\\mathrm d\\xi+\\int^\\infty_{-\\infty} cae^{-b\\xi^2}\\mathrm d\\xi\\end{eqnarray*}$$
\n\u3068\u306a\u308b\u304c\u3001\u7b2c1\u9805\u306f\u5947\u95a2\u6570\u3068\u5076\u95a2\u6570\u306e\u7a4d\u306a\u306e\u3067\u7a4d\u5206\u5024\u306f\\(0\\)\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u7b2c2\u9805\u306e\u307f\u8a08\u7b97\u3059\u308c\u3070\u3088\u304f\u3001\u3053\u308c\u306f\u516c\u5f0f\u304b\u3089\u76f4\u3061\u306b\\(\\displaystyle ca\\sqrt{\\frac \\pi b }\\) \u3068\u306a\u308b\u3051\u308c\u3069\u3001(2)\u5f0f\u3088\u308a\\(\\displaystyle a\\sqrt{\\frac \\pi b }=1\\) \u306a\u306e\u3067\u3001\u7d50\u5c40\\(\\mu=c\\) \u3067\u3042\u308b\u3002
\n\u3000\u3088\u3063\u3066\u3001\u5206\u6563\u306f$$\\sigma^2=\\int^\\infty_{-\\infty} \\xi^2ae^{-b\\xi^2}\\mathrm d\\xi$$\u3068\u306a\u308b\u306e\u3067\u3001\u90e8\u5206\u7a4d\u5206\u3057\u3066\u3044\u304f\u3068
\n$$\\begin{eqnarray*}\\sigma^2&=&\\int^\\infty_{-\\infty} \\xi\\cdot\\xi ae^{-b\\xi^2}\\mathrm d\\xi\\\\
\n&=&\\left[-\\frac{\\xi}{2b} ae^{-b\\xi^2}\\right]^\\infty_{-\\infty}+\\int^\\infty_{-\\infty} \\frac{1}{2b}ae^{-b\\xi^2}\\mathrm d\\xi\\end{eqnarray*}$$\u3068\u306a\u308b\u306e\u3060\u304c\u3001\u3053\u3053\u3067\u7b2c1\u9805\u304c\\(0\\)\u306b\u53ce\u675f\u3059\u308b\u306e\u306f\u81ea\u660e\u3067\u826f\u3044\u3060\u308d\u3046\u3002\u3088\u3063\u3066\u7d50\u5c40\u7b2c2\u9805\u306e\u307f\u304c\u6b8b\u3063\u3066\u3001\u3053\u308c\u306f\u516c\u5f0f\u304b\u3089\u76f4\u3061\u306b\\(\\displaystyle\\frac 1{2b}a\\sqrt{\\frac \\pi b}\\)\u3067\u3042\u308b\u3002
\n\u3000\u3053\u3053\u3067\u307e\u305f(2)\u5f0f\u3092\u4f7f\u3048\u3070\u3001\\(\\displaystyle\\sigma^2=\\frac 1 {2b}\\Leftrightarrow b=\\frac 1 {2\\sigma^2}\\)\u304c\u5f97\u3089\u308c\u3001\u3086\u3048\u306b\\(\\displaystyle a=\\frac 1 {\\sigma\\sqrt{2\\pi}}\\)\u3068\u306a\u308b\u3002<\/p>\n
\u3000\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\u3001\u671f\u5f85\u5024\u304c\\(\\mu\\)\u3067\u5206\u6563\u304c\\(\\sigma^2\\)\u3067\u3042\u308b\u3088\u3046\u306a\u6b63\u898f\u5206\u5e03\u306e\u78ba\u7387\u5bc6\u5ea6\u95a2\u6570\u304c$$f(x)=\\frac 1{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}$$\u3067\u8868\u305b\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u3000\u81ea\u5206\u7528\u5099\u5fd8\u9332 \u3000 \u30fb\u30ac\u30a6\u30b9\u95a2\u6570 $$f(x)=ae^{-b(x-c)^2} (a,b,c\\in \\mathbb R)$$\u306e\u5f62\u3092\u3057\u305f\u95a2\u6570\u3092\u30ac\u30a6\u30b9\u95a2\u6570\u3068\u3044\u3046\u3089\u3057\u3044\u3002 \\(x-c=\\xi\\) \u3068\u7f6e\u304d\u63db\u3048\u308c\u3070\u3001\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[83],"tags":[66,65],"_links":{"self":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/777"}],"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=777"}],"version-history":[{"count":10,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/777\/revisions"}],"predecessor-version":[{"id":787,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/posts\/777\/revisions\/787"}],"wp:attachment":[{"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/media?parent=777"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/categories?post=777"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mell0w-5phere.net\/jaded5phere\/wp-json\/wp\/v2\/tags?post=777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}