Question

In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be μtually independent exponentially distributed random variables with respective means of 1.0, 1.5, and 2.4. What is the probability that losses exceed a certain value for each event?

A. (e^-1), (e^-1.5), (e^-2.4)
B. (1 - e^-1), (1 - e^-1.5), (1 - e^-2.4)
C. (1 - e^1), (1 - e^1.5), (1 - e^2.4)
D. (e^1), (e^1.5), (e^2.4)

Answer 1

The probability that losses exceed a certain value for storm, fire, and theft, given their respective means, is calculated using the exponential distribution CDF. The probabilities are (e^{-1}), (e^{-1.5}), and (e^{-2.4}), which corresponds to answer choice A.

Step-by-step explanation:

The exponential distribution is often used to model the time between independent events that happen at a continuous but unpredictable rate. Given the exponentially distributed random variables with the means provided for storm (μ=1.0), fire (μ=1.5), and theft (μ=2.4) losses, the probability that losses exceed a certain value for each event can be calculated using the cumulative distribution function (CDF) of the exponential distribution, P(X ≥ x) = 1 - P(X ≤ x) = 1 - (1 - e-mx).

Since the question seems to refer to these variables exceeding their respective means, we apply the formula with x equal to the mean μ for each variable:

• Probability for storm losses: P(T > 1) = 1 - (1 - e-1⋅1) = e-1
• Probability for fire losses: P(T > 1.5) = 1 - (1 - e-1.5⋅1.5) = e-1.5
• Probability for theft losses: P(T > 2.4) = 1 - (1 - e-2.4⋅2.4) = e-2.4

Therefore, the correct answer is A. (e-1), (e-1.5), (e-2.4).