Q3 (25%) In an agricultural business, the profitability (P) in any given year is modeled utilizing the
income (I), labor costs (L) and material costs (M) during the same year, all measured in $/mĀ². Given
I, L, and M are normally distributed random variables with mean of 40, 25, and 10 $/mĀ² and standard
deviations 4, 3, and 2 $/mĀ², respectively. Accordingly, by using four significant decimal digits (e.g.
0.1234):
(6%) a) Find the probability that profit will be less than 10 $/mĀ² assuming income, labor cost and
material costs are all statistically independent.
(4%) b) Repeat part (a) assuming that the correlation coefficient between labor and material costs is
0.50 while income and costs are statistically independent.
(4%) c) Find the probability that this business will make less than 10 $/mĀ² profit in four out of five
years assuming all variables are statistically independent. Assume statistical independence between
the income and the costs among years.
(4%) d) Find the probability that this business will not be able to make profit higher than 10 $/mĀ² in
any of the years during a five-year period. Assume statistical independence between the income and
the costs among years.
(3%) e) Define the return period (in a single sentence, max 15 words).
(4%) f) Calculate the return period for the profit to exceed 20. Assume statistical independence
between the income and the costs among years.
