Moment

Topics: Probability - Expected Value

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As an Expected Value §

Central §

(definition)

Let $X$ be a random variable and let $\mathbb{E}[X]$ be its expected value.

We say that the central moment of $X$ of order $r$ is:

$$m_r=\mathbb{E}[(X-\mathbb{E}[X]{)}^r]$$

The second moment is better known as variance; the third is known as skewness; the fourth is known as kurtosis.

Developed Form

Recall that, from the law of the unconscious statistician, for $X$ a discrete random variable such that $\mathbb{E}[X]=\mu$:

$$\mathbb{E}[(X-\mathbb{E}[X]{)}^r]=\sum _{R_{X^r}}(x-\mu {)}^k{f}_X(x)$$

And for a similar $X$, an absolutely continuous random variable:

$$\mathbb{E}[(X-\mathbb{E}[X]{)}^r]={\int }_{\mathbb{R}}(x-\mu {)}^k{f}_X(x)dx$$

Non-Central §

(theorem)

We say that the (non-central) moment of $X$ of order $r$ is:

$$m_r^{\prime }=\mathbb{E}[X^r]$$

The Expected Values of the Powers of $X$

Notice that the expected value of $X$ is its first non-central moment.

More generally, notice that the non-central moments of $X$ are the expected values of the powers of $X$.

Developed Form

Recall that, from the law of the unconscious statistician, for $X$ a discrete random variable:

$$\mathbb{E}[X^r]=\sum _{R_{X^r}}{x}^r{f}_X(x)$$

And for $X$ an absolutely continuous random variable:

$$\mathbb{E}[X^r]={\int }_{\mathbb{R}}{x}^r{f}_X(x)dx$$

(observation)

Notice that when $\mathbb{E}[X]=0$, then $m_k=m_k^{\prime }\forall k\in \mathbb{N}$.

Properties §

(theorem)

Let $X$ be a random variable with a moment of finite order $n$. Then:

• $\mathbb{E}[X^r]$ exists for $r\le n$
• $\mathbb{E}[(X-a{)}^r]$ exists for each $r\le n$ and for every $a\in \mathbb{R}$

As a Measure of Shape §

Central §

(definition)

Let $x_1,\dots ,x_n$ be observations of a variable $X$.

The $k$th central moment of $X$ (i.e. of order $k$) is defined as:

$$m_k(X)=\frac{1}{n}\sum _{i=1}^n(x_i-\overline{X}{)}^k$$

…where $\overline{X}$ denotes the mean of $X$.

Non-Central §

(definition)

On the other hand, the $k$th (non-central) moment of $X$ is:

$$m_k^{\prime }(X)=\frac{1}{n}\sum _{i=0}^n{x}_i^k$$