Moment, Expectation, Variance, Skewness and Kurtosis

0. DictionaryPermalink

1. MomentPermalink

1.1 Definition in PhysicsPermalink

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 MomentPermalink

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 ${\mu }_0^{\prime }=1$

1.3 Central MomentPermalink

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 MomentPermalink

特定地，有 ${\alpha }_1=0$ 和 ${\alpha }_2=1$

2. ExpectationPermalink

2.1 Expectation Equals Arithmetic MeanPermalink

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 OperatorPermalink

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：

• $E[X^n]={\mu }_n^{\prime }$
• $E[(X-\mu {)}^n]={\mu }_n$
• $E\left[(\frac{X-\mu }{\sigma }{)}^n\right]=\frac{E[(X-\mu {)}^n]}{{\sigma }^n}={\alpha }_n$

3. VariancePermalink

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

4. SkewnessPermalink

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. KurtosisPermalink

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]）。