## Calculating the skewness of a probability distribution

This post presents exercises on calculating the moment coefficient of skewness. These exercises are to reinforce the calculation demonstrated in this companion blog post.

For a given random variable $X$, the Pearson’s moment coefficient of skewness (or the coefficient of skewness) is denoted by $\gamma_1$ and is defined as follows:

\displaystyle \begin{aligned} \gamma_1&=\frac{E[ (X-\mu)^3 ]}{\sigma^3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) \\&=\frac{E(X^3)-3 \mu E(X^2)+3 \mu^2 E(X)-\mu^3}{\sigma^3} \\&=\frac{E(X^3)-3 \mu [E(X^2)+\mu E(X)]-\mu^3}{\sigma^3} \\&=\frac{E(X^3)-3 \mu \sigma^2-\mu^3}{\sigma^3} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2) \\&=\frac{E(X^3)-3 \mu \sigma^2-\mu^3}{(\sigma^2)^{\frac{3}{2}}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3) \end{aligned}

(1) is the definition which is the ratio of the third central moment to the cube of the standard deviation. (2) and (3) are forms that may be easier to calculate. Essentially, if the first three raw moments $E(X)$, $E(X^2)$ and $E(X^3)$ are calculated, then the skewness coefficient can be derived via (3). For a more detailed discussion, see the companion blog post.

_____________________________________________________________________________________

Practice Problems

Practice Problems 1
Let $X$ be a random variable with density function $f(x)=10 x^9$ where $0. This is a beta distribution. Calculate the moment coefficient of skewness in two ways. One is to use formula (3) above. The other is to use the following formula for the skewness coefficient for beta distribution.

$\displaystyle \gamma_1=\frac{2(\beta-\alpha) \ \sqrt{\alpha+\beta+1}}{(\alpha+\beta+2) \ \sqrt{\alpha \ \beta}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$

$\text{ }$

Practice Problems 2
Calculate the moment coefficient of skewness for $Y=X^2$ where $X$ is as in Practice Problem 1. It will be helpful to first calculate a formula for the raw moments $E(X^k)$ of $X$.

$\text{ }$

Practice Problems 3
Let $X$ be a random variable with density function $f(x)=8 (1-x)^7$ where $0. This is a beta distribution. Calculate the moment coefficient of skewness using (4).

$\text{ }$

Practice Problems 4
Suppose that $X$ follows a gamma distribution with PDF $f(x)=4 x e^{-2x}$ where $x>0$.

• Show that $E(X)=1$, $E(X^2)=\frac{3}{2}$ and $E(X^3)=3$.
• Use the first three raw moments to calculate the moment coefficient of skewness.

$\text{ }$

Practice Problems 5
Calculate the moment coefficient of skewness for $Y=X^2$ where $X$ is as in Practice Problem 4. It will be helpful to first calculate a formula for the raw moments $E(X^k)$ of $X$.

$\text{ }$

Practice Problems 6
Verify the calculation of $\gamma_1$ and the associated calculation of Example 6 in this companion blog post.

$\text{ }$

Practice Problems 7
Verify the calculation of $\gamma_1$ and the associated calculation of Example 7 in this companion blog post.

$\text{ }$

Practice Problems 8
Verify the calculation of $\gamma_1$ and the associated calculation of Example 8 in this companion blog post.

$\text{ }$
_____________________________________________________________________________________

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$
_____________________________________________________________________________________

Practice Problems 1

• $\displaystyle \gamma_1=\frac{-36 \sqrt{3}}{13 \sqrt{10}}=-1.516770159$

$\text{ }$

Practice Problems 2

• $\displaystyle \gamma_1=\frac{- \sqrt{7}}{\sqrt{5}}=-1.183215957$

$\text{ }$

Practice Problems 3

• $\displaystyle \gamma_1=\frac{7 \sqrt{10}}{11 \sqrt{2}}=1.422952349$

$\text{ }$

Practice Problems 4

• $\displaystyle \gamma_1=\sqrt{2}$

$\text{ }$

Practice Problems 5

• $\displaystyle \gamma_1=\frac{138}{7 \sqrt{21}}=4.302009836$

$\text{ }$

_____________________________________________________________________________________

$\copyright \ 2015 \text{ by Dan Ma}$