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 $f(t)=\frac{1}{3650.1} e^{-\psi 3650.1}$ 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
$f(t)=\frac{1}{3650.1} e^{-\psi 3650.1}$
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 describes the time between events in a Poisson process, where events occur independently and at a constant average rate. It is often used to model waiting times or lifetimes, and it is characterized by a single rate parameter (?), which is the reciprocal of the mean time between events.

Probability Density Function (PDF)

A probability density function (PDF) indicates the relative likelihood for a random variable to take on a given value. For the exponential distribution, the PDF is expressed as f(t) = ?e^(–?t) for t ? 0. This function shows that the probability density decreases exponentially with time, signifying that longer waiting times are progressively less likely.

Cumulative Distribution Function (CDF) and Integration

The cumulative distribution function (CDF) represents the probability that a random variable takes a value less than or equal to a specific point, and is obtained by integrating the PDF from zero up to that point. For the exponential distribution, the CDF is F(t) = 1 – e^(–?t). This formulation is useful for calculating the probability over an interval by taking the difference of CDF values at the endpoints.

Unit Conversion in Probability Calculations

Consistency in units is essential in any probability calculation. When the time is given in one unit (such as days) but the problem involves another unit (like years), converting between units accurately ensures that the parameters and computed probabilities remain consistent and valid.

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