1. Suppose we are planning to build a dike to hold back flood waters. Let the random variable Y be the highest water level in feet in any given year at the planned dike’s location. Suppose that Y ~ N(10, 4)

a. (5) If we build the dike 12 feet high, what is the probability that the water will overflow in any given year?

b. (5) What is the 99 ^th percentile of the distribution of Y ?

c. (10) If we build the dike as high as your answer from part (b), what is the probability that the water will overflow at least once in the next 100 years ? Assume that different years yield independent realizations Y ₁ , Y ₂ , Y ₃ , …etc.

2. (10) Global Positioning Systems (GPS), using satellites, can determine the latitude, longitude, and altitude of a hand-held GPS sensor on the ground with great accuracy. Suppose that the errors X, Y, and Z (in meters) in these three coordinates are independent, normally distributed random variables with mean 0 and standard deviation ½ meter. What is the probability that the overall GPS position in three-dimensional space is accurate to within 1 meter ?

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