几种常见的概率分布率.pptxVIP

第三章:几种常见的概率分布率一随机变量(randomvariable)----随机试验中被测定的量称为随机变量．观察值（observation)----随机变量所取的值．随机变量{离散型随机变量(discretrandomvariable)连续型随机变量(continuousrandomvariable)连续型随机变量(continuousrandomvariable)—取值为某一区间内有限或无限的任何值.离散型随机变量(discretrandomvariable)—取值为有限,或可数无穷个孤立的数值.

2.普阿松分布:----小概率事件(p≦0.1)符合普阿松式分布.同样:把样本看作一个整体,则:∑f(x)=1故:式中任一项出现的概率为:μxμ----平均数P(x)=e-μx!x----第x项为自然数:1,2,3,…e----常数=2.718281…3.Poisson分布的特征:1).小概率事件.P≦0.1.2).n∞,越大越好,但事实上不可能,因此,所得的分布是个近似分布.3).μ=np大小适中,恰好为e-μ.即:与自然数e的负指数为宜.4).样本平均数就是总体平均数.X=μ.5).平均数等于方差.X=δ26).偏斜度:γ1=1/√n7).峭度:γ2=1/μ4.Poisson分布的应用:例:麦田内杂草的分布:调查已知每10平方米有一株杂草.问:100平方米有0株,1株,2株,3株…10株杂草的概率?

解:100μ=10=10株杂草数(x)概率杂草数(x)概率0P(0)=e-10.100/0!=0.0000458P(8)=e-10.108/8!=0.11261P(1)=e-10.101/1!=0.0004509P(9)=e-10.109/9!=0.12512P(2)=e-10.102/2!=0.002310P(10)=e-10.1010/10!=0.12513P(3)=e-10.103/3!=0.007611P(11)=e-10.1011/11!=0.11374P(4)=e-10.104/4!=0.018912P(12)=e-10.1012/12!=0.09485P(5)=e-10.105/5!=0.037813P(13)=e-10.1013/13!=0.07296P(6)=e-10.106/6!=0.063114P(14)=e-10.1014/14!=0.05217P(7)=e-10.107/7!=0.0901≥15P(≥15)=e-10.1015/15!=0.0835

例:卢瑟福(Rutherford)物理实验:观察在7.5秒时间间隔内放射性物质放射的质粒数到达某指定区域的质点数:X=∑fx/N=10086/2608=3.8672≈3.877.5秒内质点数(x)观察频率(f)fxμxP(x)=e–μ.x!N×P(x)0570P(0)=e-3.87×3.870/0!=0.020954.50721203203P(0)=e-3.87×3.871/1!=0.0807210.46562283766P(0)=e-3.87×3.872/2!=0.1562407.369635251575P(0)=e-3.87×3.873/3!=0.2015525.512045322128P(0)=e-3.87×3.874/4!=0.1949508.299254082040P(0)=e-3.87×3.875/5!=0.1509393.547262731638P(0)=e-3.87×3.876/6!=0.0973253.75847139973P(0)=e-3.87×3.877/7!=0.0538140.3104845360P(0)=e-3.87×3.878/8!=0.026067.8080927243P(0)=e-3.87×3.879/9!=0.011229.2096≥1016160P(0)=e-3.87×3.8710/10!=0.006617.2128∑N=26081008610000

五、超几何分布:CxnCn-xN-KP(x)=CnN