二项分布
二项分布n重伯努利试验满足下列条件:
(1)一次试验只有两种可能结果,“成功”“失败”
(2)一次试验“成功”的概率为p,“失败”的概率为 q=1-p
(3)试验相互独立
(4)试验可重复进行n次
(5)在n次试验中,“成功”的次数对应一个离散型随机变量。
在n次试验中,出现“成功”的次数的概率分布就是二项分布。
在n次试验中,出现x次成功的概率为:
data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfUAAABECAIAAAAXyzsiAAAQSUlEQVR4Ae1dK5PjOBAWXBgYuMRbGXT0gMDAg2d2cKtMDg4wD3JNLRpivsBg4ZR/QeASVw2cMhqWuzqyYECgbzbfVq9Wr/ghO47TBilFllutT+1PUuth0fDFCDACjAAjsEQExBILxWViBBgBRoARaJjf2QgYAUaAEVgmAszvy6xXLhUjwAgwAszvbAOMACPACCwTAeb3ZdYrl4oRYAQYAeZ3tgFGgBFgBJaJAPP7MuuVS8UIMAKMAPM72wAjwAgwAstEgPl9mfU6QamKosiyrCgK5LXf77Pjtd/vJ8ids2AEGIGTCDC/n4SIE1gQkFImSZJlWRRFcRxXVSWlzLIsSZIoina7neUZjmIEGIFpEWB+nxbvReQmpaRue57nQgji9CiKhBBZli2ioFwIRuCyEWB+v+z6m177LMviOKZ8i6IQQkgpERNFkZSSXTSEDwcYgTMiwPx+RvAvMmvqqkP7NE25w36RFclKXwECzO9XUMlBi0ieGUiVUgohqqoKmgkLYwQYgQAIML8HAPGaRUTH65oR4LIzArNFgPl9tlVzAYppzvcL0JhVZASuCQHm92uq7dBlTZJEc77XdV2WZeh8WB4jwAj0QWBEfn9+ft5ut9++feuj10U98/T0dH9/fw0l1aoFqyFV53uSJHmea8n4LyPACJwFgbH4/fHxcb1ef/ny5Sylmj7Tu7u7zWbz1qRNn3W/HKuqwu5TbS2jf2tSURTY0NQ0zX6/15zviOmnDz/FCDACwRH4we/7/V56ryRJ/G++qtl2u12tVq70WZaZWak7YuI41hLUda3Kn2f44eFhtVo9Pj7OUz1otdvt4jgGL8vjFUVRkiS4iz2oCFt/sVoG6bFVNYoiaiGklNx5t+K2yMiiKPCqxnGsraoKWN48z9M0DSgQooqiSJJEHq8syy6CYXqA8LP/XlVVnucYcWONc3W8drsdGFkIoW5scWW23W6FEC5yb5qmrmuAK45XFEV5nr++vpJAdBKxKzJNU48oemQmgZNlP6Oe+/0+jmNUogopmvbkeJ1c6ZimaRRFaZri3ajrGhuaYCFqIz1SScuyTJIEhJKmKY5JGJgX5BRFQcJDEUpd19A2TVOwYRzHqqkP1Pxcj9d1LaWMoqgoiqqq8MJKKcOyJPoi6u65UOWVxwvHJWEOSQhBvZxQucxBzk9+hzae/eW45Ufh69evQoiPHz+2KRsEWtsMtVPZRtRM0hwOh81ms16v5+aLL8sSaFv7Wa+vr7gbRdFJJOu6Vo8Va5qmKIo8z6kXf1JC7wToajRNU1UV+hA4AKe3wKZpdrsdDkoDW0FUFEVBqIpsO4qiLMuqqgoleUiRhz8LclfnXWBCtI15SBY0LsQoMzi/m32C3W7n4aIhZTn7s7/wO14bVycOOxU9FHA4HN6/f9+e3dCdVAf4gAN9ybND008BtHB//fVXv8fHeKosSwyVrOSOHFG5Qd7PMYoAmdS3QIcxSEZZlmFkQ8IxKDGbq5eXl533enp6UlXK8xzjJExLBGkwVPnnCmdZJoQwhzg4iciM76pnXddVVQEu+AMDmqV2ugbphkIJITzvCCW+oMAv/I5CuhgcdCyEcFnq3d2dEOLz588ty7/b7cA7qtMWA/+WEuaZ7OPHj0KImTjiwVZCCBVkEzfUxQQOFjPrHjFxHBMd93jcfETtvH/69MlKKNvt9tZ7/fnnn6bkpmng9rTeusTIm5sbaxcQvcObm5uAhQrO7xh5mMZT1zW4yFr1AUs0sahf+N2Ppn8IfzgcVqvVu3fvDodD+zJAJmFaFAWF2wuZW0pw5R9//DEHxVCn5ChwqYSXUx1xu1LOIV6l4+HubLhNqFxSyk+fPjVNM1wyZKqtkTksoHwvIgDb3mw2Vm03m41/7s36lCfSz0ieB123QDjW9snPby6BM4//hd9RQmsnjto3F1N8/vxZCNHVLwG3AMYEGHS7Bgczx1FTb71eCyFeXl60+In/0qhTnVC16qBxnDXN2SMxDw9VQZTF8RqoGCaHIYScM8PXVJRlCdipNdrv92bPcaDyEz/uH+J7CKSfnsH5HQKtToiF87vf+Y5ZZtNXTtV2e3srhOi64J2aDUyqzK3/WNc1rSaEfwONENaAe5qiv//+Wwix3W4Jn7ME2pssVndMrCTmtbAgBwurCG2zkwFar6qKTGW325309oK7ac0+vkOiFVNKSXIwggxCxFgYBucM5p/7kXtvI9SKGeQv8YBVGuzN1QW0PuKPDM7v+MqYucUacyTB53K10sF0pZQgurIs4S8i+9TSD//7s//uaZlxzEgURS7+PRwO8F79888/XXWiFrXfzMZbH/mNRvtdDw8PHm3BBeiCYe4oOS6YfX19xV+PHX/58kUI8fvvv3vkq7feJi36FWG73WrTeiQWtWadCqM05w3AomB4WKSoom1SPBa67I+XtobHLAjW8NFh9Kg764ukzUzkx8sU2DUGq32oRFouLaUNMcI3w+htVK5ZNLytrik68HtAF2twfnfBTi+LaXWuR7rG4+tmTdNgSTEMHsNQeL08fNI1L0r/k98JyuOq9x8/6NZh0bTHdfj8/CyEePfuHcltH2jvQ7DKHGLE9/f3ntkCKSVeTuSr+hZhx57uGFbRrNdrq85m5FtL0/tVVJVUJZ+cD1cTTx/O8xwGTR0LtfcQHa8hWqFnpBotaq0fzw7RZMizQ4zwzTB6G5Wr67NUfif2G1JZ/mdpaay1FRxu8Nbcf/I7ckXbIpUrTVOPIwJC0f68f//emoc/EsjObX8BsQ+Uh/Pqt99+w9+yLP008fLyggGNv+yj3qUKHTWX3sJps6t5SBn6ONZJsJbZeWSeNOaWWUyQbKARjqHhIvnd2sMIix68i+qpHpodjsvv5Hx3dQb9pcXk6u3trT+ZeRdL3WkhvJngXDFxHKtOOvLPtNcH/H6uKdaw/sQhwwuXBwmek//++w/tkGnuvfkdkzraXBEAcTkW2lfrlCmHG2FwbZfH72DesVe+0ykLcARprhj0jwP6tajef/TfqQWjG50C/fgdLlFsgAQb9nPBd1K1X2K0QJ3UOy+/U4PtcSIRFGmaqo0ZxVNgDH5vmubff/+10i4pr3psSJmTAXTetbfF+l6dFDWrBD2MMLj+C+N3bLvVPjkZHDRVIIxTG/1jGSFN8qvpB4Z/8DuqTXsl2ovu4Z/RlrqjE6c1a+0VGDslRk+qM9ef49n9M7Qwqc18EXkG/YUa466Vdgf2NmBL2ttifa/GKNF4Mrsa4Ria+IfaQL43jZgKD+QlU6AWg0maft0ITVTLv9bRKraMaUPYlgL9yX7wu/WV8D+p3u06v2oudcfrp42p1Sxc4fHmVylHdCc7De3nML+KOj3J71pDS6WeJmClXbzV/Rr7/X6vzoRTKazvFd2df6CHEY4xv+pveluaXHu0R+V3HH6p7WIbleuto1V0xcLu+yWEv/M7DYf7Od+bpum0PhIOLw1HjAC0jwGRlp7AeOsjKVPYtOrowNo7SmAG5rA+suW7oa3QMMsyaoyVdhHZzxphzDc3N+pgy/pejVqu4MJ7GOGQro9rfSTGWx8+fLAW8MOHD2Ed2S1t2KqMP5Kcw2oynKWqxoQNW0erqnPmJLF01ec7v/vb5JYSW+5vArlbHdl4qzt1k1vq1iOZuggaRqbqnCSJv188h/1NMCb/kChJEs2P0QOr3o9Yadcksk7yaXJVferkfgU18cTh+ngep3X+Y6ARjlSQ3ufPlGXZdVdwJ37Pskzza7sQsJI7DhN1OZfyPLe+8q54qzLW0araxUmPl0vtHvHf+b0TiK482pxPgP2KLkKhswr6ddxcivWIBx2gpQEHqWs50ET5xc7kfALUrDryUNU++zmdZgsEtF3vmKq8JwynKiVAfZ08YY3STxygbo064MCpZPgEAi2q62qEIxUEJGW+xf6GmSaEOlVue2rCxECb0QOcw2maZsYVx7FZLnUBiPYqwYDNtd2AyFRGpXLUDk3w4q+/N9ajQr+f84mVHjii2tpGtZHrOV9M/UIITFZrZtGFQeGRwNr6tVEjSBrVUkGCdB49mihrb4uyhq9pbueLqfQBn7vVlKkUEwTwGhC29BExVdUeauA8ZNgYHe1tPXKkh/Dgj5DZay/FQCMMricJJEpSq4kiNXc2PUX83r4i6OjTNl8OIX73T9sQI4P0zF+tFqA/Onwmj7viSRl10G8drdJ21qZpxhhMC60N0xooqp42Adf5wNhWrmakMUtZlmZzqqVpo0DANPSRATR4dFJEHMcnp7lndT4w/G9YeoHNa9ijf7IUAcF0iQK1VVWFY2dOfh3QJceMV48JolNuzGRziMHxMnEcm12rIUY4atFwaoKUEt9vwttBx6q4si7LEhWtzb1p6bMsI3M1A55n0RXzjw+oNTWZHTEu+bSAXdPWFZ+mqdZUgN/NWiZbNW9pefX4+2P9TI8nzUe6ft/DlLCAmBl+3wOo4rMJLvOdHnlXdya4Juiy+bt1wTPtKhC9n65PnTe9+v1Vjcs8ip1sBlwjAI9MumU9P47uYi1J5b3UxAsIh+T3pmnAbi2/z7cA+LQizPb7fJqec/g7Ge1aJ7XmgICqQxzH82l6VcWCh0ddQOGa7QxeiksRGJjf34o9529Mj10r11z2rthORrsYks/BH+WCaOxlea58p48vimKIB/ikwnRc6MmUV5IgPL+/Abfdbler1dmXwUxchQ8PD6vVaiaf5Zu47F2zw3ScEMI/U91VrJmenO9DRv2m2LAx2jkzYYXPR9rr6+uo/Jtl2Xkn7eYDNWkyCr83TfP4+Lher7t+7oPUurjA3d3dZrN5fn6+OM17KEyfP6Zn4dKkv/6AOWk2RueaPhWiZudX7Cx36Vjws+Q+ZaZxHKvrScJmvdvt/DOrYbO7FGlj8XvTNG9kt91uv337dilY9Nbz6enp/v7+GkraNA12ddOiRiy1lMeLPkTXG0l+kBFgBAIiMCK/B9SSRc0EAfqMH+2cklKS6wM95ZmoymowAowA8zvbQAcEsOTZtecZpH8l60A6oMZJGYEzIcD8fibgLzNb2j+9yDMaL7NOWGtGwIkA87sTGr6hIbDf77E+wXqi6WT7lTSt+C8jwAi4EGB+dyHD8U4EcOyGtpB5sv1KTrX4BiPACPyKAPP7r3jwvxYI4Pgkba2bNbKFME7CCDACYyHA/D4WsguWi3Uy6vGBOAaPtp6PsZ59wXhy0RiBkRBgfh8J2MWKtX5AQ3POENEvFgUuGCNwCQgwv19CLc1JR6vzXXXO0Br5OWnNujAC14gA8/s11vqQMqtUTnLUTyad94OupBIHGAFGgPmdbaAbAlbnO75EUVUVvrHQTSKnZgQYgXEQYH4fB9flSsWXtszypWlKu1vNuxzDCDAC0yPA/D495pwjI8AIMAJTIMD8PgXKnAcjwAgwAtMjwPw+PeacIyPACDACUyDA/D4FypwHI8AIMALTI8D8Pj3mnCMjwAgwAlMgwPw+BcqcByPACDAC0yPA/D495pwjI8AIMAJTIMD8PgXKnAcjwAgwAtMjwPw+PeacIyPACDACUyDA/D4FypwHI8AIMALTI/A/b4JLHorIir4AAAAASUVORK5CYII=
称随机变量X服从参数为(n,p)的二项分布, 记作 X~B(n,p)。
μ=E(X)=np V(X)=var(x)=npq=np(1-p) , data:image/png;base64,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
当抽样的样本量小于有限总体其个体总数的10%时, 二项分布可以作为超几何分布的近似。
二项分布的参数n足够大( 比如超过100),参数 p 不是太大或太小( 0.1 < p <0.9),二项分布 B(n,p) 可以用正态分布 N(np,np(1-p)) 近似。
谢谢分享 谢谢分享 {:1_180:}
谢谢分享 {:1_180:} {:1_180:} {:1_180:}{:1_180:}{:1_180:}{:1_180:}{:1_180:}{:1_180:} 谢谢分享 谢谢分享
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