Earthquakes The time between major earthquakes in the Taiwan...

Earthquakes The time between major earthquakes in the Taiwan region is a random variable with probability density function $$f(t)=\frac{1}{3650.1} e^{-t / 3650.1} $$ where $t$ is measured in days. Source: Journal of Seismology. a. Find the probability that the time between a major earthquake and the next one is more than 1 year but less than 3 years. b. Find the probability that the time between a major earthquake and the next one is more than 7300 days.

Earthquakes The time between major earthquakes in the Taiwan region is a random variable with probability density function
$$f(t)=\frac{1}{3650.1} e^{-t / 3650.1} $$
where $t$ is measured in days. Source: Journal of Seismology.
a. Find the probability that the time between a major earthquake and the next one is more than 1 year but less than 3 years.
b. Find the probability that the time between a major earthquake and the next one is more than 7300 days.

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model the time between independent events that occur at a constant average rate. It is characterized by a single parameter, usually represented as ? (or its reciprocal, the mean), and has the memoryless property, meaning that future probabilities are independent of past events.

Probability Density Function

The probability density function (PDF) of the exponential distribution is given by f(t) = (1/?) e^(?t/?), where ? is the mean or scale parameter. This function provides the likelihood of the random variable taking on a specific value and is fundamental in calculating probabilities over intervals by integrating the PDF.

Cumulative Distribution Function

The cumulative distribution function (CDF) of the exponential distribution, which is derived from the PDF, represents the probability that the random variable is less than or equal to a particular value. It is expressed as F(t) = 1 ? e^(?t/?). This function is used to compute probabilities for a given range by evaluating the difference between CDF values at the endpoints of that range.

Survival Function

The survival function, or tail probability, is complementary to the CDF and gives the probability that the random variable exceeds a certain value; it is defined as S(t) = 1 ? F(t) = e^(?t/?). This concept is particularly useful in reliability analysis and in assessing the probability of an event occurring after a specified time period.

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