# Moment, Expectation, Variance, Skewness and Kurtosis

## 矩、期望、方差、偏度与峰度

Yao Yao on September 4, 2014

ToC:

## 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

$\mu'_n = \int^\infty_{-\infty} (x-c)^n f(x) dx$

### 1.2 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. $c = 0$).

$\mu'_n = \int^\infty_{-\infty} x^n f(x) dx$

Of a discrete random variable $X$:

$\mu'_n = \sum_{i=1}^k x_i^n P(X = x_i)$

### 1.3 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):

$\mu_n = \int^\infty_{-\infty} (x-\mu)^n f(x) dx$

### 1.4 Standardized Moment

$\alpha_n = \frac{\mu_n}{\sigma^n}$

## 2. Expectation

### 2.1 Expectation Equals Arithmetic Mean

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

$\mu = \mu'_1 = \int^\infty_{-\infty} x f(x) dx$

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

$E[X] = \mu = \mu'_1 = \int^\infty_{-\infty} x f(x) dx$

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

$E[Y] = E[g(X)] = \int^\infty_{-\infty} g(x) f(x) dx$

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

## 3. Variance

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

$\sigma^2 = \mu_2 = \int^\infty_{-\infty} (x-\mu)^2 f(x) dx = E[(X-\mu)^2] = E[X^2] - \mu^2$

## 4. Skewness

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

$\gamma_1 = \alpha_3 = \frac{\mu_3}{\sigma^3}$

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 • 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

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:

$\gamma_2 = \alpha_4 - 3 = \frac{\mu_4}{\sigma^4} - 3$

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]）。