# Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution

### 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.

## Joint Distribution and Joint PMF

### Examples

**Directions:**There are no examples for this video.

## Marginal PMF

### Examples

**Directions:**There are no examples for this video.

## Joint PMF of Continuous Random Variables

### Examples

**Directions:**There are no examples for this video.

## Joint PDF

### There are no conceptual videos for this topic.

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

- Verify \(f\) is a joint probability density function.
- Find \(P(X<Y).\)
- Find \(E\left[X^2Y\right].\)
- Find the marginal pmf for \(X.\)

## Joint PMF and Marginal PMF

### There are no conceptual videos for this topic.

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

- Find the joint pmf of \(X_1,\) \(X_2.\)
- Find the joint pmf of \(Y_1,\) \(Y_2.\)
- Find the marginal pmf's of \(Y_1,\) \(Y_2.\)

### 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 videos listed above on this page.

- What does a joint distribution mean? How is it related to vector-valued random variables?
- Define the joint probability mass function of two discrete random variables.
- Can we find the joint distribution of two random variables knowing their individual distribution? What about the converse question? Consider both discrete and continuous cases.

## Uniform Distribution

### Examples

**Directions:**You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.

### 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.

- How do you find the marginal distributions of the coordinates of a point randomly selected inside a solid of finite volume?