Application: Sketchy Distributions

We have seen several examples which involved sketching plots of joint, conditional, and marginal distributions; including:

In this application you will create and solve your own problem of this type. The problems do not necessarily need to be more difficult than the ones we have seen in class. However, you are strongly encouraged to be creative!

The problems that you construct should be your own creation, and your solutions should be your own work. You are NOT allowed to copy material from outside sources. You can consult sources, but you must CITE any sources that you consult and include references (e.g., to justify a numerical value of a probability in context). In particular, you can NOT just copy an exercise or its solution from a textbook or other online source. To repeat: The problems that you construct should be your own creation, and your solutions should be your own work.

  1. Create your problem set up, following these specifications
  1. You must have two main continuous random variables \(X\) and \(Y\).
  2. \(X\) and \(Y\) must not be independent.
  3. You can include any other discrete or continuous random variables you want, and these can be independent of \(X\) and \(Y\) (like \(U\) in Example 20.4).
  4. Provide an enough information to determine the joint distribution of \(X\) and \(Y\). You do not have to directly specify the joint distribution, but you have to provide enough information to specify it indirectly. There are three main ways to do this:
    • Define \(Y\) as a transformation of \(X\) and other random variables, like in Example 20.4. (Note: you can’t just define \(Y\) as a transformation of \(X\) because then they would be completely dependent.)
    • Define the marginal distribution of \(X\) and then the family of conditional distributions of \(Y\) given values of \(X\), like in PP: Conditional Distributions.
    • Define the joint distribution of \(X\) and \(Y\) directly, like in Example 23.3. This is probability the hardest method.
  5. However you define your set up, you can use Uniform, Normal, and Exponential distributions in specifying marginal or conditional distributions.
  6. You must not just change numbers from previous examples. You can use previous examples as guides, but you are creating your own example.
  7. You must not use an example from another textbook or source. You can use other examples as guides, but you are creating your own example. If you consult any sources outside of course materials posted in Canvas you must cite your sources.
  1. Invent a context for your example: what do the variables \(X\) and \(Y\) represent and why does your setup make sense in this context? It doesn’t have to be entirely realistic but make sure your variables at least have reasonable scales in context. (Don’t say “heights follow a Uniform(-10, 10) distribution”.) Pick a context that is interesting to you personally and be creative!

  2. Once you have your setup, you will sketch by hand plots of each of the following, in whichever order makes the most sense given your setup.

  1. The joint distribution of \(X\) and \(Y\).

  2. At least one conditional distribution of \(Y\) given a value of \(X\).

  3. At least one conditional distribution of \(X\) given a value of \(Y\).

  4. The marginal distribution of \(X\).

  5. The marginal distribution of \(Y\).

    Your variable axes should be clearly labeled with appropriate values. Don’t worry about labeling density axes (unless it’s easy, e.g., for Uniform distributions). You should sketch plots by hand. They don’t need to be perfect, but they should exhibit the most important features. For the joint distribution plot, represent density as best as you can with shading.

    For each plot, write a few sentences describing the process you went through in constructing your plot. Also describe in words the most important features of the distribution the plot represents.

    Note: you are not just plugging formulas into plotting software to create plots. You need to describe intuitively how you get from the setup of your problem to the main features in the plots without using software.

  1. Run a simulation and create a plot of each of the distributions you sketched by hand.

    Of course, there is nothing stopping you from doing this part before sketching your plots. But your explanations of your sketches should clearly describe how you get from your setup to the plots. You should not attempt to reverse engineer your explanations based on your simulation results.

    There is a template Colab notebook that you can use, but you’ll need to fill in all of the details. For example code, see the examples mentioned above and other examples in the textbook.