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.
1. Let \(X, Y\) be random variables with a joint probability density function \[f(x,y)=\begin{cases} \frac{3}{2}\left(xy^2+y\right), & 0\leq x,y\leq 1\\ 0, & \text{otherwise}\end{cases}\]
- 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.
1. We roll two 4-sided fair dice numbered with 1, 2, 3, 4. Let \(X_1\), \(X_2\) be the outcomes, and \(Y_1=\min\left(X_1,X_2\right),\) \(Y_2=\left|X_1-X_2\right|.\)
- 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.
1. We select a random point \((X,Y)\) inside the triangle \(D\) uniformly. Find the pdf for \(X\) and \(Y.\)
![Image of a triangle with vertices (0,1), (1,0), and (2,0)](/getattachment/f3ee7f04-3412-454e-ba95-6e02f48f9a24/diagram-20230102.png)
2. Find the expected area of the rectangle with vertices \((0,0),\) \((0,X),\) \((X,Y),\) and \((Y,0)\), where the point \((X,Y)\) is selected uniformly inside the triangle in the previous example.
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?