goldnumberone 发表于 2025-7-4 08:06:36

正态分布

正态分布
如果随机变量X的概率密度函数为:
data:image/png;base64,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
则称X为正态随机变量,或称服从参数为μ, σ2的正态分布,记作 X~N(μ,σ2) 。
正态分布的概率密度函数f(x)具有下述特点:
(1)曲线的图形是一个单峰钟型曲线,它是关于直线x= μ对称的;
(2)曲线在x=μ处达到最高点,从这个最高点出发,向正负两个方向下降,无限逼近横轴(x轴) ,这条曲线与横轴质检的面积等于1。而且,曲线下在μ-σ与μ+σ之间的面积为0.6826,在μ-2σ与μ+2σ之间的面积为0.9545,在μ-3σ与μ+3σ之间的面积为0.9973。
(3)正态分布由参数μ和σ完全确定。μ反映了正态分布的中心位置和相应随机变量取值的集中位置。σ反映了分布的分散程度。
2、标准正态分布
μ=0, σ=1的正态分布称为标准正态分布
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
标准正态分布的μ=0 ,σ= 1。

马宝儿 发表于 2025-7-4 08:21:03

学到了

wuyue323 发表于 2025-7-4 09:58:17

学习了

pipahuang 发表于 2025-7-4 11:52:12

{:1_180:}

TBD16888 发表于 2025-7-4 14:44:09

谢谢分享

品质小老弟 发表于 2025-7-4 14:56:48

{:1_180:}

jzcyt2008 发表于 2025-7-4 15:21:59

{:1_101:}

flyerchang 发表于 2025-7-5 06:55:08

谢谢分享

ひろみ 发表于 2025-7-5 08:36:56

变更管理时可以用用
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查看完整版本: 正态分布