Answer :
In a 4-year period, expect around 2 loads (Poisson model). Over 11 loads is likely (80% chance). No loads for 5.8 years or more guarantees under 30% chance.
In a Poisson process with a mean time between occurrences of λ = 0.4 years, we can answer your questions as follows:
**(a) Expected number of loads in 4 years:**
The expected number of events in a Poisson process is the product of the mean occurrence rate (λ) and the time interval (t). Therefore, in a 4-year period, you can expect:
Expected loads = λ * t = 0.4 years/load * 4 years ≈ 1.6 loads
However, since the Poisson process deals with discrete events, the expected value doesn't necessarily translate to an exact number of occurrences. It represents the average number of loads you would expect to see over many repetitions of the same 4-year period.
**(b) Probability of more than 11 loads in 4 years:**
To calculate the probability of more than 11 loads, we need to find the probability of 12 or more loads. In a Poisson process, the probability of k events occurring in a time interval t is given by:
P(k events) = (e^(-λt) * (λt)^k) / k!
For k = 12 and λt = 4 * 0.4 = 1.6:
P(more than 11 loads) = 1 - P(0 loads) - P(1 load) - ... - P(11 loads)
≈ 1 - (e^(-1.6) * 1.6^0) / 0! - (e^(-1.6) * 1.6^1) / 1! - ... - (e^(-1.6) * 1.6^11) / 11!
≈ 1 - 0.1995 ≈ 0.8005
Therefore, the probability of having more than 11 loads in a 4-year period is approximately 0.801 (rounded to three decimal places).
**(c) Time period for at most 0.3 probability of no loads:**
We want to find the time period (t) where the probability of no loads (P(0 loads)) is at most 0.3. In other words, we need to solve:
P(0 loads) = e^(-λt) ≤ 0.3
Taking the natural logarithm of both sides and rearranging:
t ≥ -ln(0.3) / λ ≈ 2.303 / 0.4 ≈ 5.76 years
Therefore, the time period must be at least 5.76 years (rounded to four decimal places) for the probability of no loads to be at most 0.3.
The probable question may be:
An article suggests that a Poisson process can be used to represent the occurrence of structural loads over time. Suppose the mean time between occurrences of loads is 0.4 year.
(a) How many loads can be expected to occur during a 4-year period?
(b) What is the probability that more than eleven loads occur during a 4-year period? (Round your answer to three decimal places.)
(c) How long must a time period be so that the probability of no loads occurring during that period is at most 0.3? (Round your answer to four decimal places.)
