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In a Poisson distribution, the expected value is a key concept that measures the central tendency, or average number of events, that you can expect in a given time period. It is denoted by \( \lambda \), which is the product of the rate of occurrence \( \alpha \) and the length of the time period \( t \).
For example, if small aircraft arrive at a rate of 8 per hour, the expected value of arrivals in a 90-minute period (which is 1.5 hours) would be \( \lambda = 8 \times 1.5 = 12 \).

This means, on average, you can expect 12 aircraft to land in that period. The expected value is particularly useful because it provides a straightforward interpretation of what's typically anticipated, helping airport staff plan for resources like runway availability and ground crew allocation.

Standard deviation is another important measure in statistics that tells us how much the number of arrivals can vary from the expected value. In a Poisson distribution, the standard deviation is simply the square root of the expected value, \( \sqrt{\lambda} \).
For the 90-minute period example where the expected value is 12, the standard deviation would be \( \sqrt{12} \approx 3.464 \). This means that the actual number of aircraft arrivals is likely to fluctuate around 12 by about 3 to 4 aircraft.

Understanding standard deviation is crucial for risk assessment. It gives a sense of reliability and predictability to the expected outcomes. Smaller standard deviations imply more predictability and less variance from the expected value.

Calculating probabilities in a Poisson distribution involves using the Poisson formula: \( P(X = k) = \frac{{\lambda^k e^{-\lambda}}}{k!} \), where \( k \) is the number of events (such as aircraft arrivals).
For instance, to find the probability of exactly 6 aircraft arriving in a 1-hour period (where \( \lambda = 8 \)), we plug these values into the formula: \( P(X = 6) = \frac{8^6 e^{-8}}{6!} \approx 0.122 \).

Moreover, knowing how to calculate probability for scenarios such as "at least" or "at most" a certain number of arrivals often involves cumulative probabilities, where values are summed across a range.

The complement rule is a fundamental principle in probability that can simplify calculations. It states that the probability of an event happening is one minus the probability of it not happening: \( P(A) = 1 - P(A^c) \). In the context of Poisson distributions, it's very helpful for calculating probabilities like "at least".
For example, to find the probability that at least 6 aircraft arrive \( P(X \geq 6) \), you can calculate it as \( 1 - P(X < 6) \). By finding the sum of probabilities for 0 to 5 arrivals, you can use the complement rule to quickly obtain \( P(X \geq 6) \).

This method streamlines many probability calculations, making it easier and more efficient, especially when dealing with distributions that extend over a range of possible outcomes.