Earthquakes The time between major earthquakes in the...

Earthquakes The time between major earthquakes in the Southerm California region is a random variable with probability density function 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 less than 365 days. (b) Find the probability that the time between a major earthquake and the next one is more than 960 days.

Earthquakes The time between major earthquakes in the Southerm California region is a random variable with probability density function
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 less than 365 days.
(b) Find the probability that the time between a major earthquake and the next one is more than 960 days.

Key Concepts

Integration in Continuous Distributions

Integration is the essential mathematical tool used in continuous probability distributions to accumulate the probabilities over an interval. It allows one to compute the likelihood that a random variable falls within a particular range by summing (integrating) the infinitesimal probabilities provided by the pdf.

Complementary Probability

Complementary probability involves calculating the probability of the event not occurring by subtracting the probability of the event from one. This concept is especially useful when the probability of the complementary event is easier to compute, and it is widely applied in scenarios involving continuous distributions.

Probability Density Function

A probability density function (pdf) is a function that describes the relative likelihood for a continuous random variable to take on a given value. It must be nonnegative for all outcomes and integrates to 1 over the entire range of possible values. The pdf is fundamental to calculating probabilities for intervals by integration.

Cumulative Distribution Function

The cumulative distribution function (cdf) represents the probability that a random variable is less than or equal to a specific value. It is obtained by integrating the probability density function from the lower bound of the variable up to that value. The cdf is useful for determining the probability of events occurring within or beyond certain thresholds.

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