# Maths, Undergraduate,Senior Year Project: Papers, Quinnipiac University

- Read the task in Box 1.
- Download each of the pdfs and read them. Keep these open so you can refer to them as you watch the video.
- Watch the video.

## The Task

**Maths, Undergraduate,Senior Year Project: Papers, Quinnipiac University**

MA 490: Senior Seminar is a one-semester required capstone course that students take in the spring semester of their senior year. This is the second year we have taught the course. Part of the catalogue description of the course requires that students do an oral and written presentation. In spring 2011 the course was taught around a theme: Stochastic Processes. Each student was assigned a problem to solve based on the course material. A large part of the solution process was choosing a model for their problem. Students gave a 20 minute oral presentation on their solution before the class and wrote a 5 – 15 paper giving the details of their solution, including giving definitions and background material, and citing any sources used. The rubric for the grade on this project was as follows: 50% correctly solving the problem; 20% on the quality of the oral presentation; 20% on the quality of the paper; 10% on their ability to generalize their solution or to make suggestions that would enhance the problem and modelling process.

Students were given an option to also produce a poster and 3 students volunteered to do so. The three posters deal with the following problems: proving the Central Limit Theorem, setting premiums for term life insurance, and a specific type of random walk. The difference between the paper and the poster is that while they were directed to write their paper at the level of a senior mathematics majors, the posters were to be accessible to a more general audience. Rewriting what you have done for a different audience is in itself a very interesting process. The students who participated found this to be a very valuable experience.

## Texts

**Life Insurance**

How does an insurance company work? Insurance of any type is all about managing risk. In this particular case, given my question, when dealing with life insurance, the insurance company attempts to manage mortality rates among its clients. The insurance company collects premiums from policy holders, invests the money (usually in low risk investments), and then reimburses this money once the person passes away or the policy matures (Investopedia). In this question we are given a life term insurance policy of 40 years starting at the age of 30 and need to find out the premium charged in order for the insurance company to turn over a profit.

Life insurance provides financial resources to your family and loved ones if you unexpectedly pass. While there is an average life expectancy for people of a specific gender or ethnicity, death caused by accident or illness is a real possibility. Any person who has financial dependents, such as children, needs to ensure that in the event of their untimely death, there will be financial assistance for their surviving dependents. Life insurance provides the financial security that can be a major help to dependants at a time of grief and uncertainty.

**Markov Chains**

When dealing with Markov chains, it is important to understand what a transition matrix is. “A Markov transition matrix is a square matrix describing the probabilities of moving from one state to another in a dynamic system. In each row are the probabilities of moving from the state represented by that row, to the other states. Thus the rows of a Markov transition matrix each add to one.” (1) All of the numbers in the transition matrix are non-negative and all of the values are between 0 and 1.

A Markov process is characterized by transitions from a given initial state to a next state. “In order to be a Markov process, the following requirements must both be satisfied:

- The probability of moving from any initial state
*i*to any next state*j*(p_{jj}) must depend only on the current state*i*. - The sum of the probabilities in any row (p
_{i1}+p_{i2}+…+p_{in}) must equal 1.0.” (4)

**Binomial Distribution**

It is imperative that one first understands the distributions given before proving that they can be approximated. The binomial distribution represents the total number of successes out of *n* Bernoulli trials under specific conditions: only two outcomes are possible on each of the n trials, the probability of success for each trial is constant, and all trials are independent of each other (Weisstein). A Bernoulli trial, which is used in the binomial distribution, is an experiment with a random result with only two possible outcomes, either a “success” or a “failure.”

From the following, *k* represents the number of success in *n* trials of a Bernoulli process with probability of success *p* (Weisstein). The binomial distribution is a discrete probability distribution that is used to analyze the possible number of times that a specific event could occur in a particular amount of trials.

Furthermore, the cumulative binomial distribution refers to a specific range of data being collected in a binomial distribution (Devore). It uses the same probability distribution as the binomial distribution, but has a specific lower limit and upper limit.

Download the transcript for this video.