术语

1 trimmed mean

修剪平均数

The trimmed mean is a family of measures of central tendency.

In contrast to the arithmetic mean , the trimmed mean is a robust measure of central tendency. For example, a small fraction of anomalousmeasurements with abnormally large deviation from the center may changethe mean value substantially. At the same time, the trimmed mean isstable in respect to presence of such abnormal extreme values, whichget "trimmed" away.

 

A trimmed mean is calculated by discarding a certain percentage of the lowest and the highest scores and then computing the mean of the remaining scores. For example, a mean trimmed 50% is computed by discarding the lower and higher 25% of the scores and taking the mean of the remaining scores.
The median is the mean trimmed 100% and the arithmetic mean is the mean trimmed 0%.
A trimmed mean is obviously less susceptible to the effects of extreme scores than is the arithmetic mean. It is therefore less susceptible to sampling fluctuation than the mean for extremely skewed distributions. It is less efficient than the mean for normal distributions.
Trimmed means are often used in Olympic scoring to minimize the effects of extreme ratings possibly caused by biased judges

 

2 Gamma distribution

（1）伽玛分布（Gamma distribution）是统计学的一种连续概率函数。Gamma分布中的参数α，称为形状参数（shape parameter），β称为尺度参数（scale parameter）。

 实验定义与观念

假设随机变量X为 等到第α件事发生所需之等候时间

概率函数

令X˜Γ(α,β)；且令： （即）。

，X > 0

其中Gamma函数之特征

Gamma积分

动差母函数、概率生成函数、期望值、变异数

Gamma分配之动差母函数m.g.f

概率生成函数 p.g.f

期望值

变异数

Gamma的加成性

当两随机变量服从Gamma分配，互相独立，且单位时间内频率相同时，Gamma分配具有加成性。

 

 

 

 

（2）伽玛分布（Gamma distribution）是统计学的一种连续机率函数。Gamma分布中的参数α，称为形状参数（shape parameter），β称为尺度参数（scale parameter）。

　　伽玛分布的期望和方差分别是:a/λ,a/λ~2或者αβ,αβ~2（其中λ=β的倒数）
　　伽玛方程表达式:Γ(x)=∫e^(-t)*t^(x-1)dt (积分的下限式0,上限式+∞)
　　利用分部积分法(integration by parts)我们可以得到
　　Γ(x)=（x-1)*Γ(x-1)
　　在概率的研究中有一个重要的分布叫做伽玛分布：
　　f(x)=λe^(-λx)(λx)^(x-1)/Γ(x) x>=0
　　=0 x<0

 

3  In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-square distribution. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-square distribution. That is, if X has the chi-square distribution with ν degrees of freedom, then according to the first definition, 1 / X has the inverse-chi-square distribution with ν degrees of freedom; while according to the second definition, ν / X has the inverse-chi-square distribution with ν degrees of freedom.

This distribution arises in Bayesian statistics.

It is a continuous distribution with a probability density function. The first definition yields a density function

    f(x; /nu) = /frac{2^{-/nu/2}}{/Gamma(/nu/2)}/,x^{-/nu/2-1} e^{-1/(2 x)}

The second definition yields a density function

    f(x; /nu) = /frac{(/nu/2)^{/nu/2}}{/Gamma(/nu/2)} x^{-/nu/2-1} e^{-/nu/(2 x)}

In both cases, x > 0 and ν is the degrees of freedom parameter. This article will deal with the first definition only. Both definitions are special cases of the scaled-inverse-chi-square distribution. For the first definition σ2 = 1 / ν and for the second definition σ2 = 1.