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 \(Y.\)
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?