Moment, Expectation, Variance, Skewness and Kurtosis

矩、期望、方差、偏度与峰度

Yao Yao on September 4, 2014

Table of content

0. Dictionary

1. Moment

2. Expectation

3. Variance

4. Skewness

5. Kurtosis


0. Dictionary

English Chinese Symbol
Moment  
$n^{th}$ n 阶  
Raw Moment 原点矩 $\mu’_n$
Central Moment 中心矩 $\mu_n$
Standardized Moment 标准矩 $\alpha_n$
Mean 平均值 $\mu$
Median 中位数  
Mode 众数  
Variance 方差 $\sigma^2$
Standard Deviation 标准差 $\sigma$
Expectation Operator 期望算子 $E[X]$
Skewness [sk’ju:nes] 偏度 $\gamma_1$
Kurtosis [kɜ:’təʊsɪs] 峰度 $\gamma_2$

1. Moment

1.1 Definition in Physics

数学中矩的概念来自于物理学。在物理学中,矩,又称动差,是用来表示物体形状的物理量。

实函数(指定义域和值域均为实数域的函数)$f(x)$ 相对于值 $c$ 的 $n$ 阶矩(the $n^{th}$ moment of a real-valued continuous function $f$ of a real variable x about a value $c$)为:

1.2 Raw Moment

主要参考 Raw Moment

In statistics, a raw moment of a univariate continuous random variable $X$ is one of a probability density function (a.k.a pdf) $f(x)$ taken about 0 (i.e. $c = 0$).

Of a discrete random variable $X$:

当 n = 1 时,它的意义就是:”$X$ 的取值 $x_i$” 乘以 “$X$ 取 $x_i$ 的概率”,然后求和。

特定地,有 $\mu’_0 = 1$

1.3 Central Moment

主要参考 Central Moment

A central moment of a univariate continuous random variable $X$ is one of a probability density function $f(x)$ taken about the mean (因为 Expectation (== Mean) 也被称为随机变量的 “中心”,所以 $c = mean(X)$ 的 moment 就被命名为 central moment):

特定地,有 $\mu_0 = 1$ 和 $\mu_1 = 0$

1.4 Standardized Moment

特定地,有 $\alpha_1 = 0$ 和 $\alpha_2 = 1$

2. Expectation

2.1 Expectation Equals Arithmetic Mean

Expectation is defined as $1^{st}$ raw moment:

Expectation is the arithmetic mean of any random variable coming from any probability distribution,这个不用怀疑,可以参见这篇 Why is expectation the same as the arithmetic mean?

2.2 Expectation Operator

其实就是把 $\mu$ 看做 a function of $x$:

If $Y = g(X)$, then:

这个 $E$ 就称为 Expectation Operator。

进而有:

  • $E[X^n] = \mu’_n$
  • $E[(X-\mu)^n] = \mu_n$
  • $E \left [ \big(\frac{X-\mu}{\sigma} \big)^n \right ] = \frac{E[(X-\mu)^n]}{\sigma^n} = \alpha_n$

3. Variance

Variance is defined as $2^{nd}$ central moment:

4. Skewness

Skewness is defined as $3^{rd}$ standardized moment:

Skewness is a measure of asymmetry [əˈsɪmɪtri]:

  • If a distribution is “pulled out” towards higher values (to the right), then it has positive skewness ($\gamma_1 > 0$,称为正偏态或右偏态).
  • If it is pulled out toward lower values, then it has negative skewness ($\gamma_1 < 0$,称为负偏态或左偏态).
  • A symmetric [sɪ’metrɪk] distribution, e.g., the Gaussian distribution, has zero skewness ($\gamma_1 = 0$).
    • 进一步还可以得到:mean == median
      • 如果是 symmetric 且是单峰分布,那么还可以得到:mean == median == mode

注意看图的时候,skewness 是个非常 confusing 的概念:

  • 左图:Negative skew ($\gamma_1 < 0$) == The distribution is skewed to the LEFT == Mean is on the left side of the peak
    • while the peak is pulled towards RIGHT
  • 右图:Positive skew ($\gamma_1 > 0$) == The distribution is skewed to the RIGHT == Mean is on the right side of the peak
    • while the peak is pulled towards LEFT

所以 skewness 最好不要根据图形去记忆,而应该根据一维坐标轴:D H@ScienceForums.Net:

One way to remember the left/right stuff is that it corresponds with the orientation of the numberline. Since negative numbers are to the left of zero, negative skewness is the same as left-skewed. The same goes for positive skewness and right-skewed.

5. Kurtosis

Kurtosis, from Greek word “kyrtos” for convex, related to word “curve”, is mainly defined by $4^{th}$ standardized moment:

It is also known as excess kurtosis (超值峰度). The “minus 3” at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero.

  • If $\gamma_2 > 0$,称为尖峰态(leptokurtic, [leptəʊ’kɜ:tɪk])
  • If $\gamma_2 < 0$,称为低峰态(platykurtic, [plæ’ti:kɜ:tɪk])。


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