[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/9387#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/9387","headline":"\u78ba\u7387 – \u751f\u6210\u95a2\u6570-Wikipedia","name":"\u78ba\u7387 – \u751f\u6210\u95a2\u6570-Wikipedia","description":"before-content-x4 \u4e00 \u78ba\u7387 – \u751f\u6210\u95a2\u6570 \u3001\u77ed\u3044 \u751f\u6210\u6a5f\u80fd [\u521d\u3081] \u307e\u305f \u751f\u6210\u6a5f\u80fd [2] \u3068\u547c\u3070\u308c\u308b\u3068\u3001\u78ba\u7387\u3067\u7406\u8ad6\u306f\u7279\u5225\u306a\u5b9f\u969b\u306e\u6a5f\u80fd\u3067\u3059\u3002\u81ea\u7136\u6570\u3078\u306e\u96e2\u6563\u78ba\u7387\u5206\u5e03\u3068\u3001\u81ea\u7136\u6570\u306b\u5024\u3092\u6301\u3064\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306f\u3001\u78ba\u7387\u3092\u751f\u6210\u3059\u308b\u6a5f\u80fd\u306b\u5272\u308a\u5f53\u3066\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u9006\u306b\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u78ba\u7387\u5206\u5e03\u307e\u305f\u306f\u5206\u5e03\u306f\u3001\u3059\u3079\u3066\u306e\u78ba\u7387\u751f\u6210\u95a2\u6570\u304b\u3089\u660e\u78ba\u306b\u518d\u69cb\u7bc9\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002 after-content-x4 \u3053\u306e\u660e\u78ba\u306a\u5272\u308a\u5f53\u3066\u306b\u57fa\u3065\u3044\u3066\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u5e03\u3068\u64cd\u4f5c\u306e\u7279\u5b9a\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u95a2\u6570\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u3068\u64cd\u4f5c\u306b\u8ee2\u9001\u3067\u304d\u308b\u3088\u3046\u306b\u3001\u751f\u6210\u6a5f\u80fd\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u78ba\u7387\u306e\u751f\u6210\u95a2\u6570\u306e\u5c0e\u51fa\u3068\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u6563\u3001\u305d\u306e\u4ed6\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u6d3e\u751f\u3068\u306e\u9593\u306b\u306f\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002\u540c\u69d8\u306b\u3001\u78ba\u304b\u306b\u72ec\u7acb\u3057\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u6dfb\u52a0\u307e\u305f\u306f\u78ba\u7387\u5206\u5e03\u306e\u6298\u308a\u7573\u307f\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306e\u4e57\u7b97\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002\u3053\u306e\u91cd\u8981\u306a\u64cd\u4f5c\u306e\u7c21\u7d20\u5316\u306b\u3088\u308a\u3001\u305f\u3068\u3048\u3070\u3001Bienaym\u00e9GaltonWatson\u30d7\u30ed\u30bb\u30b9\u306a\u3069\u306e\u8907\u96d1\u306a\u78ba\u7387\u7684\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3092\u8abf\u67fb\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u3001\u4e00\u65b9\u3067\u306f\u78ba\u7387\u5206\u5e03\u306b\u3088\u3063\u3066\u3001\u4ed6\u65b9\u3067\u306f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u5e03\u306b\u3088\u3063\u3066\u30012\u3064\u306e\u65b9\u6cd5\u3067\u6307\u5b9a\u3067\u304d\u307e\u3059\u3002\u4e21\u65b9\u306e\u7a2e\u306f\u3001\u3059\u3079\u3066\u306e\u78ba\u7387\u5206\u5e03\u304c\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u5e03\u3068\u3057\u3066\u7406\u89e3\u3067\u304d\u308b\u3068\u3044\u3046\u610f\u5473\u3067\u540c\u7b49\u3067\u3042\u308a\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u3059\u3079\u3066\u306e\u5206\u5e03\u304c\u518d\u3073\u78ba\u7387\u5206\u5e03\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002\u3069\u3061\u3089\u306e\u5b9a\u7fa9\u3067\u3082 0","datePublished":"2019-03-05","dateModified":"2019-03-05","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/873703719945b62cbc02280c289e65c1da17b1de","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/873703719945b62cbc02280c289e65c1da17b1de","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/9387","wordCount":19831,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u4e00 \u78ba\u7387 – \u751f\u6210\u95a2\u6570 \u3001\u77ed\u3044 \u751f\u6210\u6a5f\u80fd [\u521d\u3081] \u307e\u305f \u751f\u6210\u6a5f\u80fd [2] \u3068\u547c\u3070\u308c\u308b\u3068\u3001\u78ba\u7387\u3067\u7406\u8ad6\u306f\u7279\u5225\u306a\u5b9f\u969b\u306e\u6a5f\u80fd\u3067\u3059\u3002\u81ea\u7136\u6570\u3078\u306e\u96e2\u6563\u78ba\u7387\u5206\u5e03\u3068\u3001\u81ea\u7136\u6570\u306b\u5024\u3092\u6301\u3064\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306f\u3001\u78ba\u7387\u3092\u751f\u6210\u3059\u308b\u6a5f\u80fd\u306b\u5272\u308a\u5f53\u3066\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u9006\u306b\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u78ba\u7387\u5206\u5e03\u307e\u305f\u306f\u5206\u5e03\u306f\u3001\u3059\u3079\u3066\u306e\u78ba\u7387\u751f\u6210\u95a2\u6570\u304b\u3089\u660e\u78ba\u306b\u518d\u69cb\u7bc9\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u3053\u306e\u660e\u78ba\u306a\u5272\u308a\u5f53\u3066\u306b\u57fa\u3065\u3044\u3066\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u5e03\u3068\u64cd\u4f5c\u306e\u7279\u5b9a\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u3092\u95a2\u6570\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u3068\u64cd\u4f5c\u306b\u8ee2\u9001\u3067\u304d\u308b\u3088\u3046\u306b\u3001\u751f\u6210\u6a5f\u80fd\u3092\u751f\u6210\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u78ba\u7387\u306e\u751f\u6210\u95a2\u6570\u306e\u5c0e\u51fa\u3068\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u6563\u3001\u305d\u306e\u4ed6\u306e\u30e2\u30fc\u30e1\u30f3\u30c8\u306e\u6d3e\u751f\u3068\u306e\u9593\u306b\u306f\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002\u540c\u69d8\u306b\u3001\u78ba\u304b\u306b\u72ec\u7acb\u3057\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u6dfb\u52a0\u307e\u305f\u306f\u78ba\u7387\u5206\u5e03\u306e\u6298\u308a\u7573\u307f\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306e\u4e57\u7b97\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002\u3053\u306e\u91cd\u8981\u306a\u64cd\u4f5c\u306e\u7c21\u7d20\u5316\u306b\u3088\u308a\u3001\u305f\u3068\u3048\u3070\u3001Bienaym\u00e9GaltonWatson\u30d7\u30ed\u30bb\u30b9\u306a\u3069\u306e\u8907\u96d1\u306a\u78ba\u7387\u7684\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3092\u8abf\u67fb\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u3001\u4e00\u65b9\u3067\u306f\u78ba\u7387\u5206\u5e03\u306b\u3088\u3063\u3066\u3001\u4ed6\u65b9\u3067\u306f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u5e03\u306b\u3088\u3063\u3066\u30012\u3064\u306e\u65b9\u6cd5\u3067\u6307\u5b9a\u3067\u304d\u307e\u3059\u3002\u4e21\u65b9\u306e\u7a2e\u306f\u3001\u3059\u3079\u3066\u306e\u78ba\u7387\u5206\u5e03\u304c\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5206\u5e03\u3068\u3057\u3066\u7406\u89e3\u3067\u304d\u308b\u3068\u3044\u3046\u610f\u5473\u3067\u540c\u7b49\u3067\u3042\u308a\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u3059\u3079\u3066\u306e\u5206\u5e03\u304c\u518d\u3073\u78ba\u7387\u5206\u5e03\u3067\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002\u3069\u3061\u3089\u306e\u5b9a\u7fa9\u3067\u3082 0 0 \uff1a= \u521d\u3081 {displaystyle 0^{0}\uff1a= 1} \u8a2d\u5b9a\u3002\u3068        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4n 0 {displaystyle mathbb {n} _ {0}} 0\u3092\u542b\u3080\u81ea\u7136\u6570\u306e\u91cf\u304c\u547c\u3073\u51fa\u3055\u308c\u307e\u3059\u3002 Table of Contents\u78ba\u7387\u5206\u5e03\u7528 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5834\u5408 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u95a2\u6570\u3068\u3057\u3066\u306e\u30d7\u30ed\u30d1\u30c6\u30a3 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u53ef\u9006\u6027 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u6298\u308a\u305f\u305f\u307f\u3068\u5408\u8a08 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u77ac\u9593\u7684\u306a\u4e16\u4ee3 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u7dda\u5f62\u5909\u63db [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u53ce\u675f [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u78ba\u7387 – \u30e9\u30f3\u30c0\u30e0\u5408\u8a08\u306e\u751f\u6210\u95a2\u6570 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u671f\u5f85\u5024\u3001\u5206\u6563\u3001\u5171\u5206\u6563 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u78ba\u7387\u5206\u5e03\u7528 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u306f p {displaystyle p} \u30aa\u30f3\u306e\u78ba\u7387\u5206\u5e03        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\uff08 n 0 \u3001 p \uff08 n 0 \uff09\uff09 \uff09\uff09 {displaystyle\uff08mathbb {n} _ {0}\u3001{mathcal {p}}\uff08mathbb {n} _ {0}\uff09} \u78ba\u7387\u95a2\u6570 f p \uff08 k \uff09\uff09 = p \uff08 { k } \uff09\uff09 {displaystyle f_ {p}\uff08k\uff09= p\uff08{k}\uff09} \u3001\u305d\u308c\u304c\u95a2\u6570\u306e\u540d\u524d\u3067\u3059 m P\uff1a [ 0 \u3001 \u521d\u3081 ] \u2192 [ 0 \u3001 \u521d\u3081 ] {displaystyle m_ {p}\u30b3\u30ed\u30f3[0,1]\u304b\u3089[0,1]} \u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059 m P\uff08 t \uff09\uff09 = \u2211 k=0\u221ef P\uff08 k \uff09\uff09 t k{displaystyle m_ {p}\uff08t\uff09= sum _ {k = 0}^{infty} f_ {p}\uff08k\uff09t^{k}} \u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570 p {displaystyle p} \u307e\u305f\u306f\u304b\u3089 f p {displaystyle f_ {p}} \u3002 [3] \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5834\u5408 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5834\u5408 \u30d0\u30c4 {displaystyle x} \u5024\u304c\u3042\u308a\u307e\u3059 n 0 {displaystyle mathbb {n} _ {0}} \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u3067\u3059 m X\uff1a [ 0 \u3001 \u521d\u3081 ] \u2192 [ 0 \u3001 \u521d\u3081 ] {displaystyle m_ {x}\u30b3\u30ed\u30f3[0,1]\u304b\u3089[0,1]} \u304b\u3089 \u30d0\u30c4 {displaystyle x} \u307e\u305f\u306f\u304b\u3089 p \u30d0\u30c4 {displaystyle p_ {x}} as\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059 m X\uff08 t \uff09\uff09 \uff1a= m P\u2218X\u22121\uff08 t \uff09\uff09 = \u2211 k=0\u221et kp [ \u30d0\u30c4 = k ] {displaystyle m_ {x}\uff08t\uff09\uff1a= m_ {pcirc x^{ –  1}}\uff08t\uff09= sum _ {k = 0}^{infty} t^{k} p [x = k]} \u3002 [4] \u3057\u305f\u304c\u3063\u3066\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u3001\u305d\u306e\u5206\u5e03\u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u3067\u3059\u3002\u3042\u308b\u3044\u306f\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u78ba\u7387\u3092\u751f\u6210\u3059\u308b\u95a2\u6570\u306f\u3001\u671f\u5f85\u5024\u3068\u3057\u3066\u306e\u671f\u5f85\u5024\u3092\u4f7f\u7528\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002 m X\uff08 t \uff09\uff09 \uff1a= \u3068 \u2061 [ tX] {displaystyle m_ {x}\uff08t\uff09\uff1a= operatorname {e}\u5de6[t^{x}\u53f3]} \u3002 [4] Bernoulli\u306b\u306f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u304c\u5206\u5e03\u3057\u3066\u3044\u307e\u3059 \u30d0\u30c4 {displaystyle x} \u3001 \u307e\u305f \u30d0\u30c4 \u301c \u65b9\u5411 \u2061 \uff08 p \uff09\uff09 {displaystyle xsim operatorname {ber}\uff08p\uff09} \u3002\u305d\u308c\u304b\u3089 p \uff08 \u30d0\u30c4 = 0 \uff09\uff09 = \u521d\u3081  –  p {displaystyle P\uff08x = 0\uff09= 1-p} \u3068 p \uff08 \u30d0\u30c4 = \u521d\u3081 \uff09\uff09 = p {displaystyle p\uff08x = 1\uff09= p} \u3002\u6b63\u5f0f\u306b\u306f\u3001\u8981\u7d04\u3057\u307e\u3059 \u30d0\u30c4 {displaystyle x} \u5168\u4f53\u306b\u5024\u3092\u6301\u3064\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3068\u3057\u3066 n 0 {displaystyle mathbb {n} _ {0}} \u30aa\u30f3\u3068\u30bb\u30c3\u30c8 p \uff08 \u30d0\u30c4 = n \uff09\uff09 = 0 {displaystyle p\uff08x = n\uff09= 0} \u305f\u3081\u306b n \u2265 2 {displaystyle ngeq 2} \u3002\u305d\u308c\u304b\u3089 m X\uff08 t \uff09\uff09 = \u2211 k=0\u221et kp [ \u30d0\u30c4 = k ] = \u521d\u3081  –  p + p t {displaystyle m_ {x}\uff08t\uff09= sum _ {k = 0}^{infty} t^{k} p [x = k] = 1-p+pt} \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3067\u3059 \u3068 {displaystyle y} \u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u3067\u5206\u5e03\u3057\u305f\u4e8c\u9805 n {displaystyle n} \u3068 p {displaystyle p} \u3001 \u307e\u305f \u3068 \u301c \u7f6e\u304d\u5834 n \u3001 p {displaystyle ysim operatorname {bin} _ {n\u3001p}} \u305d\u3046\u3067\u3059 k \u2264 n {displaystyle kleq n}  p \uff08 \u30d0\u30c4 = k \uff09\uff09 = (nk)p k\uff08 \u521d\u3081  –  p \uff09\uff09 n\u2212k{displaystyle p\uff08x = k\uff09= {binom {n} {k}} p^{k}\uff081-p\uff09^{n-k}} \u3068 p \uff08 \u30d0\u30c4 = k \uff09\uff09 = 0 {displaystyle p\uff08x = k\uff09= 0} \u305f\u3081\u306b  n “>\u3002\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059 m X\uff08 t \uff09\uff09 = \u2211 k=0n(nk)\uff08 p t \uff09\uff09 k\uff08 \u521d\u3081  –  p \uff09\uff09 n\u2212k= \uff08 p t + \u521d\u3081  –  p \uff09\uff09 nmm Slavetle State StateyState\u00f3tState\u2014 Happ\uff09MumM\u00f3eMKome Hyom Hym 1  –  Opup\uff0911-5\uff09Mup\uff09Mupe\uff09Mupe hupm hupm hupm 12-22-2 \u3002 \u3053\u308c\u306f\u3001\u30d3\u30ce\u30df\u30a2\u30f3\u6559\u80b2\u7387\u3092\u4f7f\u7528\u3057\u3066\u7d9a\u304d\u307e\u3059\u3002 \u95a2\u6570\u3068\u3057\u3066\u306e\u30d7\u30ed\u30d1\u30c6\u30a3 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u30d1\u30ef\u30fc\u30b7\u30ea\u30fc\u30ba\u3067\u3042\u308a\u3001\u5c11\u306a\u304f\u3068\u30821\u306e\u53ce\u675f\u534a\u5f84\u3092\u6301\u3063\u3066\u3044\u307e\u3059\u3001\u305d\u308c\u306f\u3059\u3079\u3066\u306e\u4eba\u306e\u305f\u3081\u306b\u53ce\u675f\u3057\u307e\u3059 t \u2208 [ 0 \u3001 \u521d\u3081 ] {displaystyletin [0,1]} \u3002\u3053\u308c\u306f\u3001\u52b9\u529b\u30b7\u30ea\u30fc\u30ba\u306e\u3059\u3079\u3066\u306e\u4fc2\u6570\u304c\u6b63\u3067\u3042\u308a\u30011\u306b\u306a\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u304b\u3089\u7d9a\u304d\u307e\u3059\u3002\u6b21\u306b\u7d9a\u304d\u307e\u3059 \u2211 k = 0 \u221e | tkp [ \u30d0\u30c4 = k ] | \u2264 \u521d\u3081 {displaystyle sum _ {k = 0}^{infty}\u5de6| t^{k} p [x = k]\u53f3| leq 1} \u305f\u3081\u306b t \u2208 [  –  \u521d\u3081 \u3001 \u521d\u3081 ] {displaystyletin [-1,1]} \u3002\u3053\u306e\u3088\u3046\u306b\u3001\u691c\u67fb\u3055\u308c\u305f\u9593\u9694\u7d99\u627f\u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f [ 0 \u3001 \u521d\u3081 ] {displaystyle [0.1]} \u52b9\u529b\u30b7\u30ea\u30fc\u30ba\u306e\u3059\u3079\u3066\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\uff1a\u305d\u308c\u3089\u306f\u5b89\u5b9a\u3057\u3066\u304a\u308a\u3001\u9593\u9694\u3067 [ 0 \u3001 \u521d\u3081 \uff09\uff09 {displaystyle [0.1\uff09} \u7121\u9650\u306b\u533a\u5225\u3055\u308c\u307e\u3059\u3002 \u305d\u308c\u305e\u308c\u306e\u30e2\u30ce\u30de\u4ee5\u6765 \u30d0\u30c4 k {displaystyle x^{k}} \u51f8\u72b6\u3068\u5358\u8abf\u6027\u304c\u6210\u9577\u3057\u3066\u304a\u308a\u3001\u3053\u308c\u3089\u306e\u7279\u6027\u306f\u5186\u9310\u5f62\u306e\u7d44\u307f\u5408\u308f\u305b\u306e\u89b3\u70b9\u304b\u3089\u5b8c\u4e86\u3057\u3001\u51f8\u3068\u5358\u8abf\u6027\u306e\u78ba\u7387\u3092\u751f\u6210\u3059\u308b\u6a5f\u80fd\u3082\u6210\u9577\u3057\u3066\u3044\u307e\u3059\u3002 \u53ef\u9006\u6027 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u3001\u306e\u5206\u5e03\u3092\u6c7a\u5b9a\u3057\u307e\u3059 \u30d0\u30c4 {displaystyle x} \u660e\u3089\u304b\u306b\uff1a \u305d\u308c\u306f \u30d0\u30c4 {displaystyle x} \u3068 \u3068 {displaystyle y}  N0{displaystyle mathbb {n} _ {0}} -\u3067\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3092\u6b69\u3044\u3066\u3044\u307e\u3059 m X\uff08 t \uff09\uff09 = m Y\uff08 t \uff09\uff09 {displaystyle m_ {x}\uff08t\uff09= m_ {y}\uff08t\uff09} \u3059\u3079\u3066\u306e\u305f\u3081\u306b t \u2208 [ 0 \u3001 c ] {displaystyletin [0\u3001c]} \u3068\u3068\u3082\u306b  0″>\u3001\u6b21\u306b\u7d9a\u304d\u307e\u3059 p [ \u30d0\u30c4 = k ] = p [ \u3068 = k ] {displaystyle p [x = k] = p [y = k]} \u3059\u3079\u3066\u306e\u305f\u3081\u306b k \u2208 N0{displaystyle kin mathbb {n} _ {0}} \u3002 \u30c6\u30a4\u30e9\u30fc\u30d5\u30a9\u30fc\u30df\u30e5\u30e9\u306e\u5f8c\u3001\u305d\u308c\u306f\u3059\u3079\u3066\u306e\u4eba\u306b\u9069\u7528\u3055\u308c\u307e\u3059 k \u2208 n 0 {displaystyle kin mathbb {n} _ {0}}  p [ \u30d0\u30c4 = k ] = mX(k)(0)k!= mY(k)(0)k!= p [ \u3068 = k ] {displaysStyle p [x = k] = {dfrac {m_ {x} {9\uff080\uff09} {m_ {m_illerial \u3002 \u3053\u306e\u63a5\u7d9a\u306f\u305d\u308c\u3092\u793a\u3057\u3066\u3044\u307e\u3059 m \u30d0\u30c4 {displaystyle m_ {x}} \u78ba\u7387 p [ \u30d0\u30c4 = k ] {displaystyle p [x = k]} \u300c\u4f5c\u6210\u300d\u3068\u78ba\u7387\u95a2\u6570\u306f\u3001\u78ba\u7387\u751f\u6210\u95a2\u6570\u304b\u3089\u518d\u69cb\u7bc9\u3067\u304d\u307e\u3059\u3002 \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u6298\u308a\u305f\u305f\u307f\u3068\u5408\u8a08 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u305d\u308c\u306f \u30d0\u30c4 {displaystyle x} \u3068 \u3068 {displaystyle y} \u72ec\u7acb n 0 {displaystyle mathbb {n} _ {0}}  – \u4fa1\u5024\u306e\u3042\u308b\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306f\u3001\u78ba\u7387\u306b\u9069\u7528\u3055\u308c\u308b – \u306e\u751f\u6210\u95a2\u6570 \u30d0\u30c4 + \u3068 {displaystyle x+y}  m X+Y\uff08 t \uff09\uff09 = \u3068 \u2061 \uff08 t X+Y\uff09\uff09 = \u3068 \u2061 \uff08 t Xde t Y\uff09\uff09 = \u3068 \u2061 \uff08 t X\uff09\uff09 de \u3068 \u2061 \uff08 t Y\uff09\uff09 = m X\uff08 t \uff09\uff09 de m Y\uff08 t \uff09\uff09 {displaystyle m_ {x+y}\uff08t\uff09= operatorname {e}\uff08t^{x+y}\uff09= operatorname {e}\uff08t^{x} cdot t^{y}\uff09= operatorname {e}\uff08t^{x}\uff09cdot operatorname {e}\uff08y}\uff08t^{x}\uff08t^}\uff09 y}\uff08t\uff09} \u3001 \u3068 \u30d0\u30c4 {displaystyle x} \u3068 \u3068 {displaystyle y} \u305d\u3046\u3067\u3059 t \u30d0\u30c4 {displaystylet^{x}} \u3068 t \u3068 {displaystylet^{y}} \u72ec\u7acb\u3002\u3053\u308c\u306f\u3001\u72ec\u7acb\u3057\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u6709\u9650\u5408\u8a08\u306b\u76f4\u63a5\u4e00\u822c\u5316\u3067\u304d\u307e\u3059\uff1a \u30d0\u30c4 \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 n {displaystyle x_ {1}\u3001ldots\u3001x_ {n}} \u72ec\u7acb n 0 {displaystyle mathbb {n} _ {0}}  – \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3092\u6b69\u3044\u3066\u304b\u3089\u9069\u7528\u3057\u307e\u3059 s n = \u2211 \u79c1 = \u521d\u3081 n \u30d0\u30c4 \u79c1 {displaystyle s_ {n} = sum _ {i = 1}^{n} x_ {i}}  m Sn\uff08 t \uff09\uff09 = \u220f i=1nm Xi\uff08 t \uff09\uff09 {displaystyle m_ {s_ {n}}\uff08t\uff09= prod _ {i = 1}^{n} m_ {x_ {i}}\uff08t\uff09}} \u3002 \u6b21\u306b\u3001\u6298\u308a\u7573\u307f\u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306e\u305f\u3081\u306b\u76f4\u63a5\u7d9a\u304d\u307e\u3059 p \u2217 Q {displaystyle p*q} \u78ba\u7387\u5bf8\u6cd5 p \u3001 Q {displaystyle P\u3001Q}  m P\u2217Q\uff08 t \uff09\uff09 = m P\uff08 t \uff09\uff09 m Q\uff08 t \uff09\uff09 {displaystyle m_ {p*q}\uff08t\uff09= m_ {p}\uff08t\uff09m_ {q}\uff08t\uff09} \u3002 \u4f8b  \u306a\u308c \u30d0\u30c4 \u521d\u3081 \u3001 \u30d0\u30c4 2 {displaystyle x_ {1}\u3001x_ {2}} \u72ec\u7acb\u3057\u305f\u3001Bernoulli\u306b\u5206\u914d\u3055\u308c\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u304c\u540c\u3058\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u306b p {displaystyle p} \u3002\u6b21\u306b\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u5408\u8a08\u306f\u3001\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u306b\u5206\u5e03\u3059\u308b\u4e8c\u9805\u3067\u3042\u308b\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u307e\u3059 2 {displaystyle 2} \u3068 p {displaystyle p} \u3001 \u307e\u305f \u30d0\u30c4 \u521d\u3081 + \u30d0\u30c4 2 \u301c \u7f6e\u304d\u5834 2 \u3001 p {displaystyle x_ {1}+x_ {2} sim operatorname {bin} _ {2\u3001p}} \u3002\u30bb\u30af\u30b7\u30e7\u30f3\u306e\u30bb\u30af\u30b7\u30e7\u30f3\u3067\u5c0e\u51fa\u3055\u308c\u3066\u3044\u308b\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03\u3068\u4e8c\u9805\u5206\u5e03\u306e\u78ba\u7387\u751f\u6210\u95a2\u6570\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002 m X1\uff08 t \uff09\uff09 de m X2\uff08 t \uff09\uff09 = \uff08 \u521d\u3081  –  p + p t \uff09\uff09 2= m Bin2,p\uff08 t \uff09\uff09 = m X1+X2\uff08 t \uff09\uff09 {displaystyle m_ {x_ {1}}\uff08t\uff09cdot m_ {x2}\uff08t\uff09=\uff081-p+pt\uff09^{2} = m_ {operatorname {bin} _ {2\u3001p}}\uff08t\uff09= m_ {x_ {1}+x_ {2}}\uff08t\uff09} \u3002 \u77ac\u9593\u7684\u306a\u4e16\u4ee3 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u306e\u305f\u3081\u306b n 0 {displaystyle mathbb {n} _ {0}}  – \u30e9\u30f3\u30c0\u30e0\u5909\u6570 \u30d0\u30c4 {displaystyle x} \u3068 k \u2208 n 0 {displaystyle kin mathbb {n} _ {0}} \u306f \u3068 \u2061 [ (Xk)] = limt\u21911mX(k)(t)k!{displaystyle operatorname {e} left [{binom {x} {k}}\u53f3] = {dfrac {lim _ {tuparrow 1} m_ {x}^{\uff08k\uff09}\uff08t\uff09} {k\uff01}}}}} \u307e\u305f \u3068 \u2061 [ X(X\u22121)\u2026(X\u2212k+1)] = \u30ea\u30e0 t\u21911m X(k)\uff08 t \uff09\uff09 {displaystyle operatorname {e}\u5de6[x\uff08x-1\uff09\u30c9\u30c3\u30c8\uff08x-k+1\uff09\u53f3] = lim _ {tuparrow 1} m_ {x}^{\uff08k\uff09}\uff08t\uff09} \u3002 2\u3064\u306e\u65b9\u7a0b\u5f0f\u306e\u4e21\u5074\u304c\u6700\u7d42\u7684\u306b\u3044\u3064\u3067\u3059 \u3068 \u2061 [ \u30d0\u30c4 k] {displaystyle operatorname {e}\u5de6[x^{k}\u53f3]} \u3064\u3044\u306b\u3002 \u7279\u306b\u3001\u671f\u5f85\u5024\u3068 n 0 {displaystyle mathbb {n} _ {0}}  – \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u88c5\u7740\u306f\u3001\u78ba\u7387\u304b\u3089\u751f\u6210\u6a5f\u80fd\u304b\u3089\u6c7a\u5b9a\u3057\u307e\u3059\u3002 \u3068 \u2061 [ \u30d0\u30c4 ] = \u30ea\u30e0 t\u21911m X‘ \uff08 t \uff09\uff09 {displaystyle operatorname {e} left [xright] = lim _ {tuparrow 1} m_ {x} ‘\uff08t\uff09} \u3001 \u3060\u3063\u305f \u2061 [ \u30d0\u30c4 ] = \u3068 \u2061 [ X(X\u22121)] + \u3068 \u2061 [ \u30d0\u30c4 ]  –  \u3068 \u2061 [X]2= \u30ea\u30e0 t\u21911\uff08 mX\u2033(t)+mX\u2032(t)\u2212mX\u2032(t)2\uff09\uff09 {displaystyle operatorname {var} left [xright] = operatorname {e}\u5de6[x\uff08x-1\uff09\u53f3]+operatorname {e} left [xright] -operatorname {e}\u5de6[x^{2} = lim _ {tuparrow 1} ‘\uff08t\uff09^{2}\u53f3\uff09} \u3002 \u53ce\u675f\u534a\u5f84\u306e\u7aef\u306b\u3042\u308b\u5897\u5f37\u5217\u306e\u5206\u5316\u306f\u5fc5\u305a\u3057\u3082\u4e0e\u3048\u200b\u200b\u3089\u308c\u306a\u3044\u305f\u3081\u3001\u5de6\u7ffc\u306e\u5236\u9650\u3092\u8003\u616e\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002 \u4f8b  \u591a\u5206 \u30d0\u30c4 {displaystyle x} \u4e8c\u9805\u5206\u5e03\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3001\u3059\u306a\u308f\u3061 \u30d0\u30c4 \u301c \u7f6e\u304d\u5834 n \u3001 p {displaystyle xsim operatorname {bin} _ {n\u3001p}} \u3002\u305d\u308c\u304b\u3089 m X\uff08 t \uff09\uff09 = \uff08 p t + \u521d\u3081  –  p \uff09\uff09 n\u3001 m X‘ \uff08 t \uff09\uff09 = n p \uff08 p t + \u521d\u3081  –  p \uff09\uff09 n\u22121\u3068 m X\u300c \uff08 t \uff09\uff09 = n \uff08 n  –  \u521d\u3081 \uff09\uff09 p 2\uff08 p t + \u521d\u3081  –  p \uff09\uff09 n\u22122{displaystyle m_ {x}\uff08t\uff09=\uff08pt+1-p\uff09^{n}\u3001quad m ‘_ {x}\uff08t\uff09= np\uff08pt+1-p\uff09^{n-1} {text {und}} m’ ‘_ {x}\uff08t\uff09= n\uff08n-1\uff09p^{2}\uff08pt+1-p+}\uff08n-1}\uff08n-1\uff09 \u4e21\u65b9\u306e\u8a98\u5c0e\u4f53\u306f\u30dd\u30ea\u30ce\u30e0\u3067\u3042\u308b\u305f\u3081\u3001\u7c21\u5358\u306b\u53ef\u80fd\u3067\u3059 t = \u521d\u3081 {displaystylet = 1} \u8a55\u4fa1\u3059\u308b\u306b\u306f\u3001\u5de6\u306e\u5236\u9650\u3092\u8003\u616e\u3059\u308b\u5fc5\u8981\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u305d\u3046\u3067\u3059 m X‘ \uff08 \u521d\u3081 \uff09\uff09 = n p \u3068 m X\u300c \uff08 \u521d\u3081 \uff09\uff09 = n \uff08 n  –  \u521d\u3081 \uff09\uff09 p 2{displaystyle m ‘_ {x}\uff081\uff09= np {text {und}} m’ ‘_ {x}\uff081\uff09= n\uff08n-1\uff09p^{2}} \u3002 \u3053\u308c\u306b\u306f\u3001\u4e0a\u8a18\u306e\u7d50\u679c\u304c\u7d9a\u304d\u307e\u3059 \u3068 \u2061 \uff08 \u30d0\u30c4 \uff09\uff09 = m X‘ \uff08 \u521d\u3081 \uff09\uff09 = n p \u3001 \u3060\u3063\u305f \u2061 \uff08 \u30d0\u30c4 \uff09\uff09 = m X\u300c \uff08 \u521d\u3081 \uff09\uff09 + m X‘ \uff08 \u521d\u3081 \uff09\uff09  –  m X‘ \uff08 \u521d\u3081 \uff09\uff09 2= n p \uff08 \u521d\u3081  –  p \uff09\uff09 {displaystyle operatorname {e}\uff08x\uff09= m ‘_ {x}\uff081\uff09= np\u3001quad operatorname {var}\uff08x\uff09= m_ {x}’ ‘\uff081\uff09+m_ {x}’\uff081\uff09-m_ {x} ‘\uff081\uff09^{2} = np\uff081-p\uff09} \u3002 \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u7dda\u5f62\u5909\u63db [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u7dda\u5f62\u5909\u63db\u306f\u3001\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306b\u6b21\u306e\u3088\u3046\u306b\u3042\u308a\u307e\u3059\u3002 m aX+b\uff08 t \uff09\uff09 = t bm X\uff08 t a\uff09\uff09 {displaystyle m_ {ax+b}\uff08t\uff09= t^{b} m_ {x}\uff08t^{a}\uff09} \u3002 \u4f8b  \u306f \u30d0\u30c4 {displaystyle x} Bernoulli\u306b\u5206\u5e03\u3057\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3001\u3064\u307e\u308a \u30d0\u30c4 \u301c \u65b9\u5411 \u2061 \uff08 p \uff09\uff09 {displaystyle xsim operatorname {ber}\uff08p\uff09} \u305d\u3046\u3067\u3059 a \u3001 b \u2208 n {displaystyle a\u3001bin mathbb {n}}} \u30e9\u30f3\u30c0\u30e0\u5909\u6570 \u3068 = a \u30d0\u30c4 + b {displaystyle y = ax+b} 2\u3064\u306e\u30dd\u30a4\u30f3\u30c8\u5206\u6563 { b \u3001 a + b } {displaystyle {b\u3001a+b}} \u3002\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059 m Y\uff08 t \uff09\uff09 = m aX+b\uff08 t \uff09\uff09 = t bm X\uff08 t a\uff09\uff09 = t bde \uff08 \u521d\u3081  –  p + p t a\uff09\uff09 = \uff08 \u521d\u3081  –  p \uff09\uff09 t b+ p t a+b{displaystyle m_ {y}\uff08t\uff09= m_ {ax+b}\uff08t\uff09= t^{b} m_ {x}\uff08t^{a}\uff09= t^{b} cdot\uff081-p+pt^{a}\uff09=\uff081-p\uff09t^{b}+pt^{a+b}}} \u3002 \u53ce\u675f [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u78ba\u7387\u751f\u6210\u95a2\u6570\u306e\u53ce\u675f\u306f\u3001\u5206\u5e03\u306e\u53ce\u675f\u306b\u76f4\u63a5\u95a2\u9023\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002 \u305d\u308c\u306f \u30d0\u30c4 \u3001 \u30d0\u30c4 1\u3001 \u30d0\u30c4 2\u3001 \u30d0\u30c4 3\u3001 … {displaystyle x\u3001x_ {1}\u3001x_ {2}\u3001x_ {3}\u3001dots} \u95a2\u9023\u3059\u308b\u78ba\u7387\u3092\u6301\u3064\u30e9\u30f3\u30c0\u30e0\u5909\u6570 – \u751f\u6210\u95a2\u6570 m \u3001 m 1\u3001 m 2\u3001 m 3\u3001 … {displayStyle M\u3001M_ {1}\u3001M_ {2}\u3001M_ {3}\u3001\u30c9\u30c3\u30c8} \u3001\u305d\u308c\u304c\u5f7c\u3089\u304c\u53ce\u675f\u3059\u308b\u65b9\u6cd5\u3067\u3059 \u30d0\u30c4 n{displaystyle x_ {n}} \u6b63\u78ba\u306b\u5206\u5e03\u3059\u308b \u30d0\u30c4 {displaystyle x} \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306e\u5834\u5408 m n{displaystyle m_ {n}} \u3059\u3079\u3066\u306e\u305f\u3081\u306b t \u2208 [ 0 \u3001 e \uff09\uff09 {displaystyletin [0\u3001varepsilon\uff09} \u3068\u3068\u3082\u306b e \u2208 \uff08 0 \u3001 \u521d\u3081 \uff09\uff09 {displaystyle varepsilon in\uff080,1\uff09} \u53cd\u5bfe\u30dd\u30a4\u30f3\u30c8 m {displaystyle m} \u53ce\u675f\u3057\u307e\u3059\u3002 [5] \u30b9\u30c6\u30fc\u30c8\u30e1\u30f3\u30c8\u306f\u3001\u78ba\u7387\u5206\u5e03\u306e\u751f\u6210\u95a2\u6570\u3068\u5f31\u3044\u53ce\u675f\u306b\u3082\u9069\u7528\u3055\u308c\u307e\u3059\u3002 \u78ba\u7387 – \u30e9\u30f3\u30c0\u30e0\u5408\u8a08\u306e\u751f\u6210\u95a2\u6570 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u78ba\u7387 – \u751f\u6210\u95a2\u6570\u3092\u4f7f\u7528\u3059\u308b\u3068\u3001\u4e71\u6570\u306esummand\u3067\u7c21\u5358\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002\u305d\u308c\u306f \uff08 \u30d0\u30c4 \u79c1 \uff09\uff09 \u79c1 \u2208 N{displaystyle\uff08x_ {i}\uff09_ {iin mathbb {n}}}}} \u5024\u3092\u6301\u3064\u72ec\u7acb\u3057\u3066\u540c\u4e00\u306b\u5206\u5e03\u3057\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570 n 0 {displaystyle mathbb {n} _ {0}} \u3068 t {displaystylet} \u3082\u30461\u3064\u3001\u307f\u3093\u306a\u304b\u3089 \u30d0\u30c4 \u79c1 {displaystyle x_ {i}} \u540c\u3058\u5024\u7bc4\u56f2\u306e\u72ec\u7acb\u3057\u305f\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u3002\u6b21\u306b\u3001\u30e9\u30f3\u30c0\u30e0\u5909\u6570\u304c\u3042\u308a\u307e\u3059 \u3068 = \u2211 i=1T\u30d0\u30c4 i{displaystyle z = sum _ {i = 1}^{t} x_ {i}} \u78ba\u7387 – \u751f\u6210\u95a2\u6570 m Z\uff08 t \uff09\uff09 = m T\uff08 m X1\uff08 t \uff09\uff09 \uff09\uff09 {displaystyle m_ {z}\uff08t\uff09= m_ {t}\uff08m_ {x_ {1}}\uff08t\uff09}} \u3002 \u3053\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u306f\u3001\u305f\u3068\u3048\u3070\u3001Galton Watson\u30d7\u30ed\u30bb\u30b9\u3092\u5206\u6790\u3059\u308b\u3068\u304d\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002\u671f\u5f85\u5024\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u4e0a\u8a18\u306e\u30eb\u30fc\u30eb\u306b\u3088\u308c\u3070\u3001\u30c1\u30a7\u30fc\u30f3\u30eb\u30fc\u30eb\u306f\u9069\u7528\u3055\u308c\u307e\u3059 \u3068 \u2061 \uff08 \u3068 \uff09\uff09 = \u3068 \u2061 \uff08 t \uff09\uff09 de \u3068 \u2061 \uff08 \u30d0\u30c4 1\uff09\uff09 {displaystyle operatorname {e}\uff08z\uff09= operatorname {e}\uff08t\uff09cdot operatorname {e}\uff08x_ {1}\uff09} \u3001 \u68ee\u306e\u5f0f\u304c\u5bfe\u5fdc\u3059\u308b\u3082\u306e\u3002 \u305d\u306e\u5f8c\u3001\u5206\u6563\u304c\u9069\u7528\u3055\u308c\u307e\u3059 \u3060\u3063\u305f \u2061 \uff08 \u3068 \uff09\uff09 = \u3060\u3063\u305f \u2061 \uff08 t \uff09\uff09 \u3068 \u2061 \uff08 \u30d0\u30c4 1\uff09\uff09 2+ \u3068 \u2061 \uff08 t \uff09\uff09 \u3060\u3063\u305f \u2061 \uff08 \u30d0\u30c4 1\uff09\uff09 {displaystyle operatorname {var}\uff08z\uff09= operatorname {var}\uff08t\uff09operatorname {e}\uff08x_ {1}\uff09^{2}+operatorname {e}\uff08t\uff09operatorname {var}\uff08x_ {1}\uff09}}}\uff08t\uff09 \u3002 \u3053\u308c\u306f\u307e\u3055\u306bBlackwell Girshick\u65b9\u7a0b\u5f0f\u3067\u3059\u3002\u307e\u305f\u3001\u5206\u6563\u3068\u88fd\u54c1\u30eb\u30fc\u30eb\u3092\u6c7a\u5b9a\u3059\u308b\u305f\u3081\u306e\u4e0a\u8a18\u306e\u30eb\u30fc\u30eb\u306b\u3082\u5f93\u3044\u307e\u3059\u3002 \u306f \u30d0\u30c4 = \uff08 \u30d0\u30c4 \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 k \uff09\uff09 {displaystyle x =\uff08x_ {1}\u3001dots\u3001x_ {k}\uff09} a k {displaystyle k}  – \u5024\u3092\u6301\u3064\u6b21\u5143\u30e9\u30f3\u30c0\u30e0\u30d9\u30af\u30c8\u30eb n 0 k {displaystyle mathbb {n} _ {0}^{k}} \u3001\u3057\u305f\u304c\u3063\u3066\u3001\u78ba\u7387 – \u751f\u6210\u95a2\u6570 \u30d0\u30c4 {displaystyle x} as\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059 m X\uff08 t \uff09\uff09 \uff1a= m X\uff08 t 1\u3001 … \u3001 t k\uff09\uff09 = \u3068 \u2061 \uff08 \u220fi=1ktiXi\uff09\uff09 = \u2211 x1,\u2026,xk=0\u221ef P\uff08 \u30d0\u30c4 1\u3001 … \u3001 \u30d0\u30c4 k\uff09\uff09 t 1x1… t kxk{displaystyle m_ {x}\uff08t\uff09\uff1a= m_ {x}\uff08t_ {1}\u3001dots\u3001t_ {k}\uff09= operatorname {e} left\uff08prod _ {i = 1}^{k} t_ {i}^{x_ {i}} red\uff09 ^{infty} f_ {p}\uff08x_ {1}\u3001ldots\u3001x_ {k}\uff09t_ {1}^{x_ {1}} dots t_ {k}^{x_ {k}}}}} \u3068 f p \uff08 \u30d0\u30c4 \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 k \uff09\uff09 = p \uff08 \u30d0\u30c4 \u521d\u3081 = \u30d0\u30c4 \u521d\u3081 \u3001 … \u3001 \u30d0\u30c4 k = \u30d0\u30c4 k \uff09\uff09 {displaystyle f_ {p}\uff08x_ {1}\u3001ldots\u3001x_ {k}\uff09= p\uff08x_ {1} = x_ {1}\u3001dotsc\u3001x_ {k} = x_ {k}\uff09} \u3002 \u671f\u5f85\u5024\u3001\u5206\u6563\u3001\u5171\u5206\u6563 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] 1\u6b21\u5143\u306e\u5834\u5408\u306b\u985e\u4f3c\u3057\u3066\u3044\u307e\u3059 \u3068 \u2061 \uff08 \u30d0\u30c4 i\uff09\uff09 = \u2202mX\u2202ti\uff08 \u521d\u3081 \u3001 … \u3001 \u521d\u3081 \uff09\uff09 \u2200 \u79c1 \u2208 { \u521d\u3081 \u3001 … \u3001 k } {displaystyle operatorname {e}\uff08x_ {i}\uff09= {frac {partial m_ {x}} {partial t_ {i}}}}}}}}}}}} Quad forall iin {1\u3001dots\u3001k}}} \u3068\u3057\u3066\u3082  \u3060\u3063\u305f \u2061 \uff08 \u30d0\u30c4 i\uff09\uff09 = \u22022mX\u2202ti2\uff08 \u521d\u3081 \u3001 … \u3001 \u521d\u3081 \uff09\uff09 + \u2202mX\u2202ti\uff08 \u521d\u3081 \u3001 … \u3001 \u521d\u3081 \uff09\uff09 \uff08 1\u2212\u2202mX\u2202ti(1,\u2026,1)\uff09\uff09 \u2200 \u79c1 \u2208 { \u521d\u3081 \u3001 … \u3001 k } {displaystyle operatorname {var}\uff08x_ {i}\uff09= {frac {partial ^{2} m_ {x}} {{partial t_ {i}} ^{2}}}}}\uff08\uff081\u3001dots\u30011\uff09+{frac {frac {x} {x} {x} {patial tial tial tial tial tial tial tial tial tial tial m_ \uff081- {frac {partial m_ {x}} {partial t_ {i}}\uff081\u3001dots\u30011\uff09right\uff09quad forall iin {1\u3001dots\u3001k}} \u3068  \u2061 \uff08 \u30d0\u30c4 i\u3001 \u30d0\u30c4 j\uff09\uff09 = \u22022mX\u2202ti\u2202tj\uff08 \u521d\u3081 \u3001 … \u3001 \u521d\u3081 \uff09\uff09  –  \u2202mX\u2202ti\uff08 \u521d\u3081 \u3001 … \u3001 \u521d\u3081 \uff09\uff09 de \u2202mX\u2202tj\uff08 \u521d\u3081 \u3001 … \u3001 \u521d\u3081 \uff09\uff09 \u2200 \u79c1 \u3001 j \u2208 { \u521d\u3081 \u3001 … \u3001 k } {displaystyle operatorname {cov}\uff08x_ {i}\u3001x_ {j}\uff09= {frac {partial ^{2} m_ {x}} {partial t_ {i} partial t_ {j}}}}}\uff081\u3001\u30c9\u30c3\u30c8\u30011\uff09 1\u3001\u30c9\u30c3\u30c8\u30011\uff09cdot {frac {partial m_ {x}} {partial t_ {j}}\uff081\u3001dots\u30011\uff09quad forall i\u3001jin {1\u3001dots\u3001k}} \u4e00\u822c\u7684\u306a\u96e2\u6563\u5206\u5e03\u306e\u3044\u304f\u3064\u304b\u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u304c\u8868\u306b\u30ea\u30b9\u30c8\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u78ba\u7387 – \u3053\u3053\u306b\u30ea\u30b9\u30c8\u3055\u308c\u3066\u3044\u306a\u3044\u78ba\u7387\u5206\u5e03\u306e\u751f\u6210\u95a2\u6570\u306f\u3001\u78ba\u7387\u95a2\u6570\u306e\u305d\u308c\u305e\u308c\u306e\u8a18\u4e8b\u306b\u3042\u308a\u307e\u3059\u3002 \u7279\u306b\u3001\u4e8c\u9805\u5206\u5e03\u306f\u30d9\u30eb\u30cc\u30fc\u30ea\u5206\u5e03\u306e\u78ba\u7387\u751f\u6210\u95a2\u6570\u306en-\u30d5\u30a1\u30c1\u30f3\u7a4d\u306b\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u4e8c\u9805\u5206\u5e03\u306f\u307e\u3055\u306b\u72ec\u7acb\u3057\u305f\u30d9\u30eb\u30cc\u30fc\u30a4\u5206\u5e03\u306e\u5408\u8a08\u3067\u3042\u308b\u305f\u3081\u3067\u3059\u3002\u540c\u3058\u3053\u3068\u304c\u5e7e\u4f55\u5b66\u7684\u5206\u5e03\u3068\u8ca0\u306e\u4e8c\u9805\u5206\u5e03\u306b\u3082\u5f53\u3066\u306f\u307e\u308a\u307e\u3059\u3002 \u30e9\u30f3\u30c0\u30e0\u5909\u6570\u306e\u78ba\u7387 – \u751f\u6210\u95a2\u6570 \u30d0\u30c4 {displaystyle x} \u78ba\u7387\u95a2\u6570 p {displaystyle p} \u751f\u6210\u95a2\u6570\u306e\u7279\u6b8a\u306a\u30b1\u30fc\u30b9\u3067\u3059 a \u79c1 = p \uff08 \u79c1 \uff09\uff09 {displaystyle a_ {i} = pleft\uff08{i} right\uff09} \u305f\u3081\u306b \u79c1 \u2208 n 0 {displaystyle iin mathbb {n} _ {0}} \u3002\u78ba\u7387 – \u751f\u6210\u95a2\u6570\u306b\u52a0\u3048\u3066\u3001\u30be\u30a6\u30e0\u30b7\u306b\u306f\u4ed6\u306b\u30823\u3064\u306e\u751f\u6210\u95a2\u6570\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u306f\u3001\u96e2\u6563\u5206\u5e03\u306e\u305f\u3081\u3060\u3051\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u3060\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u30e2\u30fc\u30e1\u30f3\u30c8 – \u751f\u6210\u95a2\u6570\u306f\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059 m \u30d0\u30c4 \uff08 t \uff09\uff09 \uff1a= \u3068 \u2061 \uff08 \u305d\u3046\u3067\u3059 tX\uff09\uff09 {displaystyle m_ {x} left\uff08tright\uff09\uff1a= operatorname {e}\u5de6\uff08e^{tx}\u53f3\uff09} \u3002\u9069\u7528\u3055\u308c\u307e\u3059 m \u30d0\u30c4 \uff08 \u305d\u3046\u3067\u3059 t\uff09\uff09 = m \u30d0\u30c4 \uff08 t \uff09\uff09 {displaystyle m_ {x}\u5de6\uff08e^{t}\u53f3\uff09= m_ {x} left\uff08tright\uff09}} \u7279\u6027\u95a2\u6570\u306f\u3001\u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059 \u30d5\u30a1\u30a4 \u30d0\u30c4 \uff08 t \uff09\uff09 \uff1a= \u3068 \u2061 \uff08 \u305d\u3046\u3067\u3059 itX\uff09\uff09 {displaystyle varphi _ {x} left\uff08tright\uff09\uff1a= operatorname {e} left\uff08e^{itx}\u53f3\uff09} \u3002\u9069\u7528\u3055\u308c\u307e\u3059 m \u30d0\u30c4 \uff08 \u305d\u3046\u3067\u3059 it\uff09\uff09 = \u30d5\u30a1\u30a4 \u30d0\u30c4 \uff08 t \uff09\uff09 {displaystyle m_ {x}\u5de6\uff08e^{it}\u53f3\uff09= varphi _ {x} left\uff08tright\uff09} \u3002 \u307e\u305f\u3001\u30e2\u30fc\u30e1\u30f3\u30c8\u751f\u6210\u95a2\u6570\u306e\u5bfe\u6570\u3068\u3057\u3066\u306e\u7d2f\u7a4d\u751f\u6210\u95a2\u6570\u3082\u3042\u308a\u307e\u3059\u3002\u7d2f\u7a4d\u306e\u6982\u5ff5\u306f\u305d\u308c\u304b\u3089\u6d3e\u751f\u3057\u3066\u3044\u307e\u3059\u3002 \u30af\u30e9\u30a6\u30b9D.\u30b7\u30e5\u30df\u30c3\u30c8\uff1a \u6e2c\u5b9a\u3068\u78ba\u7387\u3002 Springer\u3001Berlin Heidelberg 2009\u3001ISBN 978-3-540-89729-3\u3001S\u3002370ff\u3002 Achim Klenke\uff1a \u78ba\u7387\u7406\u8ad6 \u3002 3.\u30a8\u30c7\u30a3\u30b7\u30e7\u30f3\u3002 Springer-Verlag\u3001Berlin Heidelberg 2013\u3001ISBN 978-3-642-36017-6\u3002  \u30a6\u30eb\u30ea\u30c3\u30d2\u30fb\u30af\u30ec\u30f3\u30b2\u30eb\uff1a \u78ba\u7387\u7406\u8ad6\u3068\u7d71\u8a08\u306e\u7d39\u4ecb \u3002\u52c9\u5f37\u3001\u5c02\u9580\u7684\u306a\u5b9f\u8df5\u3001\u6559\u80b2\u3002 8.\u30a8\u30c7\u30a3\u30b7\u30e7\u30f3\u3002 Vieweg\u3001Wiesbaden 2005\u3001ISBN 3-8348-0063-5\u3002  \u30cf\u30f3\u30b9\u30fb\u30aa\u30c3\u30c8\u30fc\u30fb\u30b8\u30e7\u30fc\u30b8\uff1a \u5b89stic \u3002\u78ba\u7387\u7406\u8ad6\u3068\u7d71\u8a08\u306e\u7d39\u4ecb\u3002\u7b2c4\u7248\u3002 Walter de Gruyter\u3001\u30d9\u30eb\u30ea\u30f32009\u3001ISBN 978-3-11-021526-7\u3002  \u30af\u30ea\u30b9\u30c1\u30e3\u30f3\u30d8\u30c3\u30bb\uff1a \u9069\u7528\u3055\u308c\u305f\u78ba\u7387\u7406\u8ad6 \u3002\u7b2c1\u7248\u3002 Vieweg\u3001Wiesbaden 2003\u3001ISBN 3-528-03183-2\u3002  \u2191  ehrhard\u306fbehrends\uff1a \u57fa\u672c\u7684\u306a\u30bc\u30ed \u3002\u5b66\u7fd2\u672c – \u5b66\u751f\u306b\u3088\u3063\u3066\u958b\u767a\u3055\u308c\u307e\u3057\u305f\u3002 Springer Spectrum\u3001Wiesbaden 2013\u3001ISBN 978-3-8348-1939-0\u3001 S.  108 \u3001doi\uff1a 10,1007\/978-3-8348-2331-1 \u3002    \u2191  Achim Klenke\uff1a \u78ba\u7387\u7406\u8ad6 \u3002 3.\u30a8\u30c7\u30a3\u30b7\u30e7\u30f3\u3002 Springer-Verlag\u3001Berlin Heidelberg 2013\u3001ISBN 978-3-642-36017-6\u3001 S.  79 \u3001doi\uff1a 10,1007\/978-3-642-36018-3 \u3002    \u2191  \u30cf\u30f3\u30b9\u30fb\u30aa\u30c3\u30c8\u30fc\u30fb\u30b8\u30e7\u30fc\u30b8\uff1a \u5b89stic \u3002\u78ba\u7387\u7406\u8ad6\u3068\u7d71\u8a08\u306e\u7d39\u4ecb\u3002\u7b2c4\u7248\u3002 Walter de Gruyter\u3001\u30d9\u30eb\u30ea\u30f32009\u3001ISBN 978-3-11-021526-7\u3001 S.  111 \u3001doi\uff1a 10.1515\/9783110215274 \u3002    \u2191 a b  \u30cf\u30f3\u30b9\u30fb\u30aa\u30c3\u30c8\u30fc\u30fb\u30b8\u30e7\u30fc\u30b8\uff1a \u5b89stic \u3002\u78ba\u7387\u7406\u8ad6\u3068\u7d71\u8a08\u306e\u7d39\u4ecb\u3002\u7b2c4\u7248\u3002 Walter de Gruyter\u3001\u30d9\u30eb\u30ea\u30f32009\u3001ISBN 978-3-11-021526-7\u3001 S.  114 \u3001doi\uff1a 10.1515\/9783110215274 \u3002    \u2191  Achim Klenke\uff1a \u78ba\u7387\u7406\u8ad6 \u3002 3.\u30a8\u30c7\u30a3\u30b7\u30e7\u30f3\u3002 Springer-Verlag\u3001Berlin Heidelberg 2013\u3001ISBN 978-3-642-36017-6\u3001 S.  83 \u3001doi\uff1a 10,1007\/978-3-642-36018-3 \u3002         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/9387#breadcrumbitem","name":"\u78ba\u7387 – 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