Statistical data of vehicular accidents show that the annual vehicle miles (i.e.

Question

Statistical data of vehicular accidents show that the annual vehicle miles (i.e. miles per vehicle per year) driven between accidents (of all severities) can be represented by a normal random variable with a mean of 15,000 miles per year and a standard deviation of 3750 miles per year. a. What is the probability that a typical driver, who drives 10,000 miles per year, has an accident in any given year? b. If the same driver has driven 8000 miles so far in a given year without encountering any accident, what is the probability of his/her having an accident for the remainder of that year? c. Now suppose we wish to consider the number of accidents per 10,000 vehicle miles, which can be modeled with a Poisson process. If the mean rate is one accident per 10,000 vehicle miles, what is the probability that a driver encounters less than two accidents after driving 60,000 miles?

Explanation / Answer

Solution:

Given that µ = 15000, = 3750

a. The respective Z-score with X = 10000 is

Z = (X - µ)/

Z = (10000 – 15000)/3750

Z = -1.33

Using Z-tables, the probability is

P [Z < -1.33] = 0.0918

b. The probability of remaining would be 10000 – 8000 = 2000

The respective Z-score with X = 2000 is

Z = (X - µ)/

Z = (2000 – 15000)/3750

Z = -3.47

Using Z-tables, the probability is

P [Z < -3.47] = 0.0003

c. We have = 60000/15000 = 4

Using Poisson process, the probability is

P [X < 2] = P (X = 0) + P (X = 1)

P [X < 2] =e^-4 (4)^0/0! + e^-4 (4)^1/1!

P (X < 2) = 0.0183 + 0.0733

P (X < 2) = 0.0916

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