Home page for accesible maths 2.6 Expectation and Related Summaries 2.6.4 Standardisation Computer Simulation

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2.6.5 Higher Moments

The pdfs shown in Figure 2.5 (First Link, Second Link), Figure 2.6 (First Link, Second Link) and Figure 2.7 (First Link, Second Link) below are for random variables $Y$ for a range of distribution functions $F_{Y}$ which all have $\operatorname{\mathsf{E}}\left[{Y}\right]=0$ and ${\operatorname{\mathsf{Var}}}\left[{Y}\right]=1$ . Despite having the same expectation and variance there are quite substantial differences between the six distributions. It is helpful to think how you would summarise such differences in the shape of the distributions.

Figure 2.5: First Link, Second Link, Caption: The pdfs for two random variables

Y

with

\operatorname{\mathsf{E}}\left[{Y}\right]=0

Figure 2.6: First Link, Second Link, Caption: The pdfs for two more random variables

Y

with

\operatorname{\mathsf{E}}\left[{Y}\right]=0

Figure 2.7: First Link, Second Link, Caption: The pdfs for another two more random variables

Y

with

\operatorname{\mathsf{E}}\left[{Y}\right]=0

The most obvious differences in the shapes are

•

skewness (lack of symmetry), and
•

pointedness (or lack thereof).

To be able to evaluate these shape characteristics we need to introduce higher moments.

The $r$ -th moment is evaluated as

\displaystyle\operatorname{\mathsf{E}}\left[{X^{r}}\right]=\left\{\begin{array% }[]{ll}\sum_{i=-\infty}^{\infty}i^{r}p_{X}(i)&\quad\text{if $X$ is discrete}\\ \int_{-\infty}^{\infty}t^{r}f_{X}(t)\,\mathrm{d}t&\quad\text{if $X$ is % continuous}\end{array}\right.

The $r$ -th standardized central moment is the expected value of the $r$ -th power of the standardized random variable $(X-\mu_{X})/\sigma_{X}$ . It is defined as

\displaystyle\mu_{r}=\operatorname{\mathsf{E}}\left[{\left(\frac{X-\mu_{X}}{% \sigma_{X}}\right)^{r}}\right].

Proposition 2.6.1.

For all odd $r\geq 3$ , any random variable which has a finite $r$ -th moment and for which the density is symmetric about the mean, $\mu$ , has an $r$ -th central moment of zero.

The proof of this result is straightforward but distracts from our main theme and so it appears in Appendix B and is non-examinable (the result itself is examinable, however).

Skewness: The third standardised central moment, $\mu_{3}$ , is a measure of the extent of the asymmetry and is called the coefficient of skewness. Positive (negative) values of the coefficient of skewness correspond to the distribution having a longer (shorter) upper tail than lower tail. If the distribution is symmetric the coefficient of skewness is zero.

Kurtosis: The extent of the pointedness of the distribution is measured by kurtosis, $\mu_{4}-3$ . The reason for subtracting $3$ is to give the Normal distribution kurtosis $0$ . The Normal distribution is studied in Chapter 3 and given in the first panel of Figure 2.5 (First Link, Second Link). Positive (negative) kurtosis correspond to the distribution being more (less) pointed than the Normal distribution.

Table 2.1 gives the third and fourth standardized central moments for the distributions in Figure 2.5 (First Link, Second Link), Figure 2.6 (First Link, Second Link) and Figure 2.7 (First Link, Second Link), but in a permuted order.

A	$\mu_{3}=0$	B	$\mu_{3}=0$	C	$\mu_{3}=0$
	$\mu_{4}=3$		$\mu_{4}=1.5$		$\mu_{4}=6$
D	$\mu_{3}=-1$	E	$\mu_{3}=0$	F	$\mu_{3}=2$
	$\mu_{4}=4.5$		$\mu_{4}=1.8$		$\mu_{4}=9$

Table 2.1: The third and fourth standardized central moments for the six distributions shown in Figure 2.5 (First Link, Second Link), Figure 2.6 (First Link, Second Link) and Figure 2.7 (First Link, Second Link) (permuted).

Example 2.6.4.

Use intuitive reasoning to match up the subsequent panels in Figure 2.5 (First Link, Second Link), Figure 2.6 (First Link, Second Link) and Figure 2.7 (First Link, Second Link) to the corresponding panels in the table.

Solution. We refer to Figure 2.5 (First Link, Second Link) as Fig1, Figure 2.6 (First Link, Second Link) as Fig2 and Figure 2.7 (First Link, Second Link) as Fig3. The symmetric ones have ${\color[rgb]{0.76,0.01,0}\mu_{3}=0}$ : [Fig1(LR), Fig2(R), Fig3(L)], [tab: A, B, C, E].

${\color[rgb]{0.76,0.01,0}\mu_{3}<0\Rightarrow}$ skew with long L tail: [Fig2(L) = tab D]. Hence [Fig3(R) = tab F].

For remaining guess ${\color[rgb]{0.76,0.01,0}\mu_{4}}$ ordering [Fig3(L), Fig1(R), Fig1(L), Fig2(R)] = [tab: B, E, A, C]. (Low ${\color[rgb]{0.76,0.01,0}\mu_{4}}$ corresponds to short tails [Fig1(R), Fig3(L)].)

Example 2.6.5.

For a random variable $X$ with pdf

\displaystyle f_{X}(x)=\left\{\begin{array}[]{ll}2x&\quad 0\leq x\leq 1\\ 0&\quad\text{otherwise}\end{array}\right.

Calculate $\operatorname{\mathsf{E}}\left[{X^{a}}\right]$ and hence calculate the skewness of the distribution.

Solution.

	$\displaystyle\operatorname{\mathsf{E}}\left[{X^{a}}\right]$	$\displaystyle=\int_{0}^{1}s^{a}\cdot 2s\,\mathrm{d}s$
		$\displaystyle=\left[\frac{2}{a+2}s^{a+2}\right]_{0}^{1}$
		$\displaystyle=\frac{2}{a+2}.$

So $\operatorname{\mathsf{E}}\left[{X}\right]=2/3$ , ${\operatorname{\mathsf{Var}}}\left[{X}\right]={\color[rgb]{0.76,0.01,0}2/4-(2/% 3)^{2}=1/18}$ and $\operatorname{\mathsf{E}}\left[{X^{3}}\right]={\color[rgb]{0.76,0.01,0}2/5.}$

Then

	$\displaystyle\mu_{3}$	$\displaystyle={\operatorname{\mathsf{E}}\left[{\left(\frac{X-\mu}{\sigma}% \right)^{3}}\right]}$
		$\displaystyle={\color[rgb]{0.76,0.01,0}\frac{1}{\sigma^{3}}\left(\operatorname% {\mathsf{E}}\left[{X^{3}}\right]-3\operatorname{\mathsf{E}}\left[{X^{2}}\right% ]\mu+3\operatorname{\mathsf{E}}\left[{X}\right]\mu^{2}-\mu^{3}\right)}$
		$\displaystyle={{2}^{3/2}3^{3}}\left(\frac{2}{5}-3\frac{1}{2}\frac{2}{3}+3\frac% {2}{3}\frac{2^{2}}{3^{2}}-\frac{2^{3}}{3^{3}}\right)$
		$\displaystyle=-\frac{{2}^{3/2}}{5}.$

Note: skewness is negative here.