Final answer:
The expectation of X is 195 minutes and the variance of X is 1792.5 minutes squared. X represents the time between each bird's arrival and follows the Exponential distribution with lambda equal to 1 / (60 minutes). The probability that the first bird arrives within 25 minutes is approximately 0.2375 and the probability that the first bird arrives between 30 minutes and 45 minutes is approximately 0.1782.
Step-by-step explanation:
1.) The expectation of X represents the average time between each bird's arrival. Since the first bird arrives after 15 minutes and birds land at a constant rate for 6 hours, we can calculate the expectation using the formula:
E(X) = 15 + (6-1) * 60 / 2 = 195 minutes
The variance of X represents the spread or variability in the time between each bird's arrival. We can calculate the variance using the formula:
Var(X) = (60^2 / 12) * (6-1) = 1792.5 minutes^2
2.) The probability distribution that would represent X is the Exponential distribution. Its parameter, lambda (λ), represents the average rate at which events occur. In this case, lambda is equal to 1 / (60 minutes) because there is 1 arrival every 60 minutes on average.
3.) To find the probability that the first bird arrives within 25 minutes, we can calculate the cumulative distribution function (CDF) of the Exponential distribution at 25 minutes:
P(X <= 25) = 1 - exp(-25 * λ) ≈ 0.2375
4.) To find the probability that the first bird arrives between 30 minutes and 45 minutes, we can calculate the difference between the CDF values at 45 minutes and 30 minutes:
P(30 <= X <= 45) = P(X <= 45) - P(X <= 30) ≈ 0.1782