（四）概率

老习惯，还是先给出该章节的思维导图让大家先有个整体的概念 对于基础概念就不在此赘述，挑其中的几个容易混淆的点和关键点说说 首先便是互斥事件与独立事件，很多人会将两者混淆。有个例子很好的说明了两者不是一回事： 如果两个事件是互斥事件，其中之一被

老习惯，还是先给出该章节的思维导图让大家先有个整体的概念

对于基础概念就不在此赘述，挑其中的几个容易混淆的点和关键点说说

首先便是互斥事件与独立事件，很多人会将两者混淆。有个例子很好的说明了两者不是一回事：

如果两个事件是互斥事件，其中之一被确定已经发生，则另一事件发生的概率降为0，显然两者是相关的。<喎?http://www.2cto.com/kf/ware/vc/" target="_blank" class="keylink">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"http://www.2cto.com/uploadfile/Collfiles/20140625/2014062509022412.png" width="300" height="200" alt="\">

性质：

正态概率分布有一个完整家族。每一特定正态分布通过其均值 μ 、标准差 σ 来区分。
正态曲线的最高点在均值，它也是分布的中位数和众数
分布的均值可以是任意数值：负数、零或正数。
正态概率分布是对称的。
曲线的尾端向两个方向无限延伸，且理论上永远不会与横轴相交。

标准差决定曲线的宽度
正态概率分布曲线下的总面积是 1，对所有的连续型概率分布都是如此。
正态随机变量的概率由曲线下面积给出。一些常用区间的概率是68.26%，95.44%，99.72%

连续修正因子：当用连续正态概率分布来近似离散二项概率分布时，从x值加减的0. 5值。

指数分布与泊松分布的关系在于，如果泊松分布给出了每一间隔中发生次数的适当描述，则指数分布可给出两次发生之间间隔长度的描述。

PS: 指数分布是偏度为2的严重右偏分布。