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

1. Read the task in Box 1.
2. Download each of the pdfs and read them. Keep these open so you can refer to them as you watch the video.
3. Watch the video. Maths, Undergraduate,Senior Year Project: Poster, 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 (Thankfully they were all able to solve their 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

### Setting Premiums for Term Life Insurance Policies

Question

Mortality rates for 30 year olds in the United States is .001; that is .1% of thirty year olds die and 99.9% live to 31. The mortality rate increases .0005 (.05%) per year. A 30 year old person buys term life insurance for \$500,000, which will expire when he/she turns 70. What should the insurance company charge per year (it stays the same) so that it expects to turn a profit over the course of time that the policy is active? Ignore costs and inflation first, and then factor in reasonable overhead costs the second time. How would you adjust for inflation?

Background

This particular problem deals with term life insurance. Term insurance gives only temporary protection. Benefits are paid only if the insured dies within the term specified in the policy. Protection and premiums cease at the end of this period. Because term insurance buys protection on a temporary basis, premiums are lower than for ordinary life insurance.

The calculations are made on a certain set of assumptions

1. Mortality rates increase linearly over time
2. The policy will terminate at age 70
3. Inflation will remain at 3% over the course of the policy
4. Income is derived slowly from premiums and not from investments

### Markov Process: Absorbing State Matrix

Transition Matrix

“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)

Markov Process

“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 (pjj) must depend only on the current state i.

- The sum of the probabilities in any row (pi1+pi2+…+pin) must equal 1.” (4)

Absorbing States

“A Markov chain is absorbing if it has at least 1 absorbing state, and if from every state it is possible to go to an absorbing state (not necessarily in one step)” (3)

“State i is absorbing if and only if pii =1 and

pij =0 for i ≠ j.” (2)

### Approximating the Cumulative Distribution and the Poisson Distribution by the Normal Distribution

Binomial Distribution:

The binomial distribution represents the total number of successes out of n Bernoulli trials where only two outcomes are possible, the probability of success for each trial is constant, and all trials are independent of each other .

Cumulative Binomial Distribution:

The cumulative binomial distribution refers to a specific range of data being collected in a binomial distribution.

The Poisson Distribution:

The Poisson distribution represents the probability of a number of times a random event occurs in a given amount of time unit.

The Normal Distribution:

The normal distribution is a probability distribution that is used to approximate continuous random variables around a single mean value.

Moment Generating Functions:

Moment generating functions redefines a specific probability distribution by using expected values of a random variable. The moment generating function of a random variable X is the function MX(t)= E(etx). The function can then be rewritten since the term etx can be approximated around zero using a Taylor series expansion. Thus the moment function becomes:  