IIIT wants to build a new academic block which will stand for a long time by protecting itagainst earthquakes. If earthquakes strike Telangana as a Poisson process with rate λ peryear and we want the building to last for Y years with probability p. What is the smallestnumber of earthquakes the new building should be able to endure? Calculate with valuesλ “ 0.1, Y “ 20, p “ 0.9
Question
IIIT wants to build a new academic block which will stand for a long time by protecting itagainst earthquakes. If earthquakes strike Telangana as a Poisson process with rate λ peryear and we want the building to last for Y years with probability p. What is the smallestnumber of earthquakes the new building should be able to endure? Calculate with valuesλ “ 0.1, Y “ 20, p “ 0.9
Solution
The number of earthquakes that occur in a given period of time can be modeled as a Poisson distribution. The Poisson distribution is defined by the rate parameter λ, which is the average number of events per unit of time.
Given that earthquakes strike Telangana as a Poisson process with rate λ = 0.1 per year, the expected number of earthquakes in Y = 20 years is λY = 0.1 * 20 = 2.
We want the building to last for Y years with probability p = 0.9. This means that the probability of having 2 or fewer earthquakes in 20 years should be 0.9.
The cumulative distribution function (CDF) of a Poisson distribution gives the probability of observing k or fewer events. So we want to find the smallest number k such that P(X ≤ k) ≥ 0.9, where X is a Poisson random variable with mean λY = 2.
Using the formula for the CDF of a Poisson distribution, we have:
P(X ≤ k) = e^-λY * Σ(i=0 to k) [(λY)^i / i!]
We want to find the smallest k such that this is greater than or equal to 0.9. This requires calculating the sum for increasing values of k until the sum is greater than or equal to 0.9.
By calculating, we find that the smallest k that satisfies this condition is 5. Therefore, the new building should be able to endure at least 5 earthquakes.
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