1 . 相互独立事件
事件A(或B)是否发生对事件B(或A)发生的概率没有影响,这样的两个事件叫作______ .
两个相互独立事件同时发生的概率![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6f2c684829b545b4d590286ec30ed0.png)
______ .
事件A(或B)是否发生对事件B(或A)发生的概率没有影响,这样的两个事件叫作
两个相互独立事件同时发生的概率
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd6f2c684829b545b4d590286ec30ed0.png)
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名校
解题方法
2 . 已知随机变量
服从正态分布
,
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/552595efb8eb89a970a1abbb0c955cca.png)
______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b204a6f61bd3e1085c3d50720c109561.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fe0171a0d4874966a49a88a702f25b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/552595efb8eb89a970a1abbb0c955cca.png)
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3 . 若某事件A发生的概率为
,则事件A在一次试验中发生的次数X的方差的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29c8578f06897aa6fb84aa95c797d3d8.png)
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4 . 正态曲线及其性质
(1)正态曲线:我们称
,
,其中
,
时为参数,为正态密度函数,称它的图象为正态密度曲线,简称正态曲线.
(2)正态分布:若随机变量X的概率分布密度函数为
,则称随机变量X服从正态分布,记为_________ .特别地,当
,
时,称随机变量X服从________ 正态分布.
(3)正态分布的期望与方差:若
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5b9249c10ae3896e2ee96bfa1a153e5.png)
______ ,
_______ .
(4)正态曲线的特点:
①非负性:对
,
,它的图象在x轴的上方.
②定值性:曲线与x轴之间的面积为1.
③对称性:曲线是单峰的,它关于直线________ 对称.
④最大值:曲线在
处达到峰值
.
⑤当
无限增大时,曲线无限接近x轴.
⑥当
一定时,曲线的位置由
确定,曲线随着
的变化而沿
轴平移,如图①.
⑦当
一定时,曲线的形状由
确定,
较小时曲线“瘦高”,表示随机变量X的分布比较集中;
较大时,曲线“矮胖”,表示随机变量X的分布比较分散,如图②.
,如图所示,X取值不超过
的概率
为图中区域A的面积,而
为区域B的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f3460bf7ce838cbfd78fabc04b6cee6.png)
.
(1)正态曲线:我们称
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab88cb5e0bf24c07c4fa50757f755dc4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1aedc1c8a16e306bcd6e5154f9ed6dfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cefeddff52b02a43831beeb4bb6faa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da01e03ec2bd0c5965c2535ab6835ef9.png)
(2)正态分布:若随机变量X的概率分布密度函数为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe7d5809da02c15a43a0e9a898b9086.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ad39768ebe8d7a3a0fd09efe7e200ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6095d3cc6af6cc00646493bcfeda216.png)
(3)正态分布的期望与方差:若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e6c106c07c72c0e54503728911c1743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5b9249c10ae3896e2ee96bfa1a153e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ab40fadc58e24cf88c93e52b289e9cc.png)
(4)正态曲线的特点:
①非负性:对
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ad5fe274cfc8da2dacfb65801f344ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/018857ec6e498113b3b12a730d9313da.png)
②定值性:曲线与x轴之间的面积为1.
③对称性:曲线是单峰的,它关于直线
④最大值:曲线在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6887eae47401f3acc6f99ce6ca7c30d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/623112afbf2e4c16f94f07f361a84bf3.png)
⑤当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe916d05211cf74a2b1428a8bb8bbbbd.png)
⑥当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
⑦当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0ad7e7853a069537387b5192f73844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e6c106c07c72c0e54503728911c1743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/073f0aebd74634327d63d5310be981d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c6b18c9e23f1b58b74f47c70667aa33.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f3460bf7ce838cbfd78fabc04b6cee6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d616424f5ab99deb3f310316e2c5171c.png)
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23-24高二下·全国·课前预习
5 . 离散型随机变量的方差
如果离散型随机变量
的分布列如表所示,
则称![](https://staticzujuan.xkw.com/quesimg/Upload/formula/708039a9eb75863d750b4e968b2302d8.png)
_____________ 为随机变量
的方差,有时也记为
,并称
为标准差,记为__________ .
在方差计算中,利用结论
经常可以使计算简化.
如果离散型随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
…… | ||||
…… |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/708039a9eb75863d750b4e968b2302d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f4c7a13b58096bc89d49550d4d1624e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ed24ed8e4ea58156c06bcfcdbdf8760.png)
在方差计算中,利用结论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e5a2bfa4a97b141cc00b33573a35459.png)
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6 . 已知甲射击命中的概率为
,且每次射击命中得
分,未命中得
分,每次射击相互独立,设甲
次射击的总得分为随机变量
,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc23dc1709498e8920d7d243213190b2.png)
__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c95b6be4554f03bf496092f1acdfbb89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07ae0b4264da6a8812454ffd2f20d94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc23dc1709498e8920d7d243213190b2.png)
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23-24高二下·全国·课前预习
7 . 超几何分布的均值
若随机变量
服从超几何分布,则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/223770da09feb2fc824764188e924d7e.png)
______ =__________ (
是
件产品的次品率).
若随机变量
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/223770da09feb2fc824764188e924d7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1010846eeec6c9da29640f5aa3f8738.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
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23-24高二下·全国·课前预习
8 . 超几何分布
一般地,假设一批产品共有
件,其中有
件次品,从
件产品中随机抽取
件(不放回),用
表示抽取的
件产品中的次品数,则
的分布列为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a80ab9c5df8c297967deee467734adb.png)
________ ,
.其中
,
,
,
,
.如果随机变量
的分布列具有上式的形式,那么称随机变量
服从超几何分布.
一般地,假设一批产品共有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a80ab9c5df8c297967deee467734adb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd1a3a87b1c4c48f74d8a3a6eb24562d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14674d8421e2131bdf2dda40dd330bb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c3e9740f1cdda4ee24958381be9117f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61265c7696f67a37ce09e2b996e81579.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8c23a6d81028f8e241e4e48c2ff3c8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b9305889b3d2b0d95575c12d3278456.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
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23-24高二下·全国·课前预习
9 . n重伯努利试验
(1)伯努利试验:我们把只包含两个可能结果的试验叫做伯努利试验;
(2)定义:将一个伯努利实验独立地重复进行
次所组成的随机试验称为n重伯努利实验;
(3)特征:(1)同一个伯努利实验重复做n次;(2)各次试验的结果______ .
(1)伯努利试验:我们把只包含两个可能结果的试验叫做伯努利试验;
(2)定义:将一个伯努利实验独立地重复进行
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
(3)特征:(1)同一个伯努利实验重复做n次;(2)各次试验的结果
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10 . 二项分布
(1)二项分布:一般地,在n重伯努利试验中,设每次试验中事件A发生的概率为
,用X表示事件A发生的次数,则X的分布列为
,
.如果随机变量X的分布列具有上式的形式,则称随机变量X服从二项分布,记作______ ,且有![](https://staticzujuan.xkw.com/quesimg/Upload/formula/223770da09feb2fc824764188e924d7e.png)
______ ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd353481279871d4c18541586c319c8d.png)
______ .
注:n次独立重复试验中恰好发生k次的概率与第k次才发生的概率计算公式分别是
与
.
(2)确定一个二项分布模型的步骤
①明确伯努利试验及事件A的意义,确定事件A发生的概率p;
②确定重复试验的次数
,并判断各次试验的独立性;
③设
的
次独立重复试验中事件
发生的次数,则
(1)二项分布:一般地,在n重伯努利试验中,设每次试验中事件A发生的概率为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44fed1be8b7e50f18cb90077d9fce8e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f9bbd3c8d5a7b42215ddb3c7bad334d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96bf724dccc4829fb895f675c37451e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/223770da09feb2fc824764188e924d7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd353481279871d4c18541586c319c8d.png)
注:n次独立重复试验中恰好发生k次的概率与第k次才发生的概率计算公式分别是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/899fa25fb87a9bac56538a29eac4a384.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4142775aca7f97f413bd3cac0194c8.png)
(2)确定一个二项分布模型的步骤
①明确伯努利试验及事件A的意义,确定事件A发生的概率p;
②确定重复试验的次数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
③设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f022950e0faa45b617d497b01b5292b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b73fc8a4163d94be65fbe933bf8edda4.png)
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