# Application: Major Distributions

In this application you will create and solve your own problems involving *each* of the following commonly used distributions.

- Binomial or Negative Binomial (pick one)
- Poisson
- Exponential
- Bivariate Normal

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.

For each of the four named distributions above, choose a context that is interesting to you. Be creative! The context doesn’t have to be entirely realistic but it should be reasonable. Explain why you chose your context and why it is reasonable to assume that your variables have the distributions you have assumed, and justify your choices for parameter values. (For example, I discussed why it might be reasonable to assume home runs in baseball games follow a Poisson distribution.) You do not have to find related data, but if you are able to, that is one way to justify your assumptions. (Be sure to cite any sources). You can have four separate contexts for each of the four problems, but you are also welcome to have a common theme for multiple problems as long as it is reasonable.

Clearly define at least two problems to solve for each context. Each problem should have a single numerical answer which should be interpreted in context. As a collection, your problems should demonstrate that you can compute and interpret

- Probability
- Conditional probability
- Expected value
- Variance or standard deviation

You do not have to do all of the above for each of the four distributions, just make sure that each of these items shows up somewhere among your four contexts.

Note: the problems do not necessarily need to be more difficult than the ones we have seen in class. You should be creative in your choice of contexts, and each problem should be interesting and relevant in context, but it’s fine if the method of solving the problem is similar to another problem we have seen. However, I encourage you to include some new features that we haven’t seen in other problems, or be very creative in the way you apply problems or features that we have already seen.

Solve the problems by hand, detailing all your steps as if you are teaching someone how to solve the problem.

**You should rely on properties of probability and distributions as much as possible, and only use calculus/algebra when absolutely necessary.**Also, justify your work using properties we have already established as much as possible, instead of deriving properties of scratch. (For example, you should use the cdf of the Exponential distribution without needing to integrate; you can use Poisson aggregation without calculating the probability of all the cases that result in a particular sum.) You can assume the person you are teaching has seen all the topics and concepts that we have; they just haven’t seen your problems before.Use Symbulate to solve your problems. You can either use built in functions like

`.cdf()`

when appropriate, or you can set up and run simulations.Of course, there is nothing stopping you from doing this part before solving your problems by hand. But your problem solutions should clearly describe how you get from your setup to the answers; just providing answers without justification is not enough.

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.