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 , the Pearson’s moment coefficient of skewness (or the coefficient of skewness) is denoted by and is defined as follows:

(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 , and 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 be a random variable with density function where . 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.

*Practice Problems 2*

Calculate the moment coefficient of skewness for where is as in Practice Problem 1. It will be helpful to first calculate a formula for the raw moments of .

*Practice Problems 3*

Let be a random variable with density function where . This is a beta distribution. Calculate the moment coefficient of skewness using (4).

*Practice Problems 4*

Suppose that follows a gamma distribution with PDF where .

- Show that , and .
- Use the first three raw moments to calculate the moment coefficient of skewness.

*Practice Problems 5*

Calculate the moment coefficient of skewness for where is as in Practice Problem 4. It will be helpful to first calculate a formula for the raw moments of .

*Practice Problems 6*

Verify the calculation of and the associated calculation of Example 6 in this companion blog post.

*Practice Problems 7*

Verify the calculation of and the associated calculation of Example 7 in this companion blog post.

*Practice Problems 8*

Verify the calculation of and the associated calculation of Example 8 in this companion blog post.

_____________________________________________________________________________________

_____________________________________________________________________________________

**Answers**

*Practice Problems 1*

*Practice Problems 2*

*Practice Problems 3*

*Practice Problems 4*

*Practice Problems 5*

_____________________________________________________________________________________