A technical discussion supported by equations, graphical results, drawings, etc. for each of the following problems. Assume that your audience is your management at work. Address each of the following problem, and don’t cut and paste from the web.
1. Simulate a constant rate process of binary events using a random number generator. Define an appropriate “interval” for counting your binary events. Then, use a binning or sorting function to compute a distribution for the data in terms of bin occupancy. Plot the corresponding theoretical expression on top of your histogram of simulated data.
2. Compute a theoretical Poisson distribution using a tree diagram for the probability of a yes being 0.1. This is a calculator or spreadsheet type exercise to demonstrate that the Poisson expression is an approximation, and that a binary tree search yields the exact distribution. Show, using the results of your computations, but not by including ridiculous lists of numbers in your submission, that the Poisson process works when pevent is small or large, but not when it is, for example, 0.4.
3. Compute, under circumstances where the Poisson distribution is accurate, a graphical comparison of the distribution with its bell-shaped Gaussian curve counterpart.
4. What is the relationship between the mean rate and the standard deviation/variance for a Poisson distribution? Does the experimental result from problem 1 support this conclusion? Why, or why not?
5. Why is it sometimes useful to use a Poisson distribution for computation purposes for processes that described more accurately by a Gaussian distribution?
6. Go to a store, fast food restaurant, etc, and gather actual data by counting the number of people in line as a function of time. Then, try to demonstrate that the data can be modelled with reasonable accuracy using a Poisson distribution.
7. Without using a random number generator, derive differential equations, and show their solutions graphically, for:
• A mortgage amortization for interest only, negative amortization, and positive amortization loans. Find an article online about payday loans and pending legislation and read it. Which of your graphs describes a payday loan?
• Using your insight on mortgages, derive, solve, and graph two or more solutions (i.e., using two sets of values for the parameters in the equation) for the situation in which a storm water management pond (or your bathtub) fills at a constant rate while draining at a constant-hazard rate. You are trying to illustrate the concept of incidence versus prevalence. Look these words up online.
• Pick the disease of your choice for which there is a known value of incidence per year. Using its corresponding prevalence, try to compute the average life expectancy of a person who has the disease. (Or, your average wait time on hold while listening to a recording that says, “Did you know that you can visit us on the web?” This is the same problem.)
• Discuss how the parameters can be adjusted to model norovirus on a ship versus norovirus on the Appalachian Trail.
• How does this relate to the AIDS epidemic? If you are interested, you might Google the name “Gaetan Dugas” and see how the Centers for Disease Control thought about this problem many years ago.
8 Find an online example of a Kaplan-Meier distribution. Relate this to the above discussion on call-center wait times or survival rates. What is the part of the incidence-prevalence problem that is ignored in a Kaplan-Meier analysis?
9. Extra credit: draw an electrical circuit using a current source, capacitor, and resistor to demonstrate the above problem(s). Simulate the behavior of the circuit if you have access to appropriate software. Explain how this technique is used to measure values of capacitance without the need for an impedance bridge. Write the transfer function of the circuit in the S-domain, and explain why the square root of negative 1 shows up with respect to payday loans.
Dear Sir,
I am a telecommunication Engineer specialized in wireless communication and signal processing. So I have good knowleadges and long experience with probability, signal processing and mathematics in general.
Best regards,
Habib Chabbi
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