MATHEMATICS DISCIPLINE

UNIVERSITY OF MINNESOTA, MORRIS

MORRIS, MN 56267

1. GESCO Insurance Company charges a $350 premium per annum for a $100,000 life insurance policy for a 40-year-old female. The probability that a 40-year-old-female will die within one year is 0.002.

Let X be a random variable that denotes the gain of the company for next year from a $100,000 life insurance policy sold to a 40-year-old female. Write the probability distribution for X.

2. Constantine et al. studied the effects of cranial radiotherapy on brain tumors in children. The distribution of hormonal abnormalities in such children is displayed in the following table (x=number of abnormalities)

a. What is the probability that these children will present more than 2 abnormalities?

b. What is the expected number of abnormalities presented by such a child?

c. If 20 such children independently selected, what is the probability that at least 5 of them will not show any abnormalities?

3. (Use Poisson Approximation to Binomial) Assume that on each day each person in a community with population 10,000 requires a hospital bed with probability 1/2000.

a. What is the probability that more than 6 persons in this community will require a hospital bed in a given day?

b. At least how many beds should the hospital have so that it can accommodate all people requiring a bed on a given day with a probability of at least .95?

4. The number of monthly breakdowns of a computer ia a random variable having a Poisson distribution with average number of breakdowns 1.8 in a month.

a. Find the probability that this computer will function for a month with at least one breakdown.

b. Given that there is at least one breakdown, what is the probability that the number of breakdowns is 3.

5. As a part of an air pollution survey, an inspector decides to examine the exhaust of six of a company’s 24 trucks. Suppose that four of the company’s trucks emit excessive amounts of pollutants.

What is the probability that none of the trucks that emit excessive amounts of pollutants will be included in the inspector’s sample?

6. Sally Ragseller has discovered that the daily sales at her clothing store are normally distributed with a mean of 5000 and a standard deviation of 500.

a. What is the probability that the daily sales will be between 4000 and 6000?

b. Sally likes to celebrate when sales are high, but she does not want to celebrate on more than 5% of the days. What is the minimum level of sales that should indicate celebration for Sally?

7. The length of time between arrivals of patients at a Hospital clinic is exponentially distributed. Suppose the mean time between arrivals for patients at a clinic is 4 minutes.

a. What is the probability that the length of time between arrivals is less than 1 minute?

b. What is the probability that the length of time between arrivals will exceed 10 minutes?

c. What is the probability that the next four interarrival times are less than 1 minute?