 # Section 8: Independence of Two Random Variables, Covariance, and Correlation

### Instructions

• This section covers the concepts listed below.
• For each concept, there is a conceptual video explaining it followed by videos working through examples.
• When you have finished the material below, you can go to the next section or return to the main Mathematical Probability page
Independence of Discrete Random Variables and Continuous Random Variables

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Define independence of random variables.
Variance of Sums and Covariance

### Examples

Directions: There is no example video for this topic.
Correlation

### Examples

Directions: The following examples cover the material from the video above.
Variance of Sums of Random Variables

### Examples

Directions: The following examples cover the material from the video above.
Properties of Covariance and the Correlation Coefficient

### Examples

Directions: There is no example video for this topic.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.

Note: These self-assessment questions apply to all the variance of sums, covariance, and correlation videos above.
1. Is variance of the sum of two random variables always equal to the sum of their variances? If not, how are they related exactly? How about their expected values?
2. How is covariance of two random variables defined? What about their correlation coefficient?
3. Is the expectation of the product of two random variables equal to the product of their expectations?
4. Are independent random variables uncorrelated? And conversely?
5. How can you use the correlation coefficient to see if they are almost linearly correlated?
Correlation between Indicator of Two Events

### Examples

Directions: There is no example video for this topic.
Applications of Indicator Random Variable

### Examples

Directions: The following examples cover the material from the video above.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. What distribution do indicator random variables follow?
2. When are two indicator random variables: positively correlated? Uncorrelated?
Exchangeable

### Examples

Directions: There is no example video for this topic.

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. When are two random variables exchangeable?
2. When are three random variables exchangeable?
3. Give an example of two exchangeable random variables.