The magnitude of earthquakes recorded in New Zealand can be modelled as having an exponential distribution with mean 2.4. Find the probability that an earthquake striking in New Zealand is larger than 3 in magnitude.
Question
The magnitude of earthquakes recorded in New Zealand can be modelled as having an exponential distribution with mean 2.4. Find the probability that an earthquake striking in New Zealand is larger than 3 in magnitude.
Solution
The exponential distribution has a probability density function (pdf) given by f(x) = (1/λ) * exp(-x/λ), where λ is the mean of the distribution.
For an exponential distribution, the probability that a random variable X is greater than some value x is given by the survival function, which is 1 - the cumulative distribution function (CDF). For an exponential distribution, the survival function is exp(-x/λ).
So, the probability that an earthquake in New Zealand is larger than 3 in magnitude is given by:
P(X > 3) = exp(-3/2.4) = exp(-1.25) = 0.2865
So, the probability that an earthquake striking in New Zealand is larger than 3 in magnitude is approximately 0.2865, or 28.65%.
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