Final answer:
The mean time between stork sightings is approximately 158.7 days. The probabilities of no sightings within 0.25, 0.5, and 3 years are approximately 32.05%, 19.47%, and 0.03%, respectively, based on the Poisson process with a mean of 2.3 sightings per year.
Step-by-step explanation:
The number of stork sightings in South Carolina is modeled by a Poisson process with a mean of 2.3 per year.
(a) Mean time between sightings
To find the mean time between sightings, take the reciprocal of the mean rate of sightings. Thus, the mean time is 1 / 2.3 years, which is approximately 0.435 years or 158.7 days between sightings.
(b) Probability of no sightings in three months
The probability of no sightings in 0.25 years (three months) can be calculated using the Poisson distribution formula P(X=k) = (e^{-λt} * (λt)^k) / k!, where λ is the average rate, t is the time interval, and k is the number of events (sightings). Since we are looking for no sightings, k = 0. The calculated probability is P(0) = e^{-2.3 * 0.25} which is approximately 0.3205, or 32.05%.
(c) Probability of first sighting exceeding six months
The probability that the first sighting exceeds six months is the same as the probability of zero sightings in six months. We can use the same formula as in part (b) with t = 0.5 years to get P(0) = e^{-2.3 * 0.5} which is about 0.1947, or 19.47%.
(d) Probability of no sighting within three years
Similarly, we can use the Poisson distribution formula with t = 3 years to find P(0) = e^{-2.3 * 3} which is approximately 0.0003, or 0.03%.