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In Poisson distribution, probability calculation is paramount. A Poisson distribution is used to predict the number of events happening within a fixed interval of time or space. The key to these calculations is the probability mass function (PMF). It is defined as follows: \( P(X = k) = \frac{e^{-\mu} \mu^k}{k!} \), where \( \mu \) is the average number of occurrences, \( k \) is the number of occurrences, and \( e \) is Euler's number, approximately 2.71828.
For example, to find the probability of exactly 6 aircraft arriving within an hour when the rate is 8, we set \( \mu = 8 \) and \( k = 6 \) to find \( P(X = 6) \). Likewise, for probabilities like "at least" something, such as "at least 6 arrivals," it involves adding probabilities from 0 to 5 and subtracting from 1 to find \( P(X \geq 6) \).
This foundational formula helps calculate precise probabilities for varied outcomes, which makes the Poisson process a powerful statistical tool.

The expected value, often denoted as \( E(X) \), in a Poisson distribution, represents the average number of occurrences expected in the interval. This is simply equal to the parameter \( \mu \) of the distribution. It is a measure of central tendency that gives an intuitive understanding of what is "average" or "normal" for the distribution.
In our problem scenario, if we need the expected number of aircraft arrivals during a 90-minute period (which is 1.5 hours), the parameter \( \mu \) becomes \( 8 \times 1.5 = 12 \). Thus, \( E(X) = 12 \), meaning on average, 12 aircraft are expected to arrive.
Understanding expected value is crucial as it guides decision making and predictions in practical scenarios.

The standard deviation in a Poisson distribution measures the variation or spread around the average number of occurrences. Importantly, for a Poisson distribution, the standard deviation \( \sigma \) is the square root of the expected value (or parameter \( \mu \)). This is expressed as \( \sigma = \sqrt{\mu} \).
For instance, during a 90-minute period with \( \mu = 12 \), the standard deviation is \( \sqrt{12} \approx 3.464 \). This indicates that while the average is 12, the number of aircraft arrivals can commonly vary by about 3.464 around this mean.
This concept helps in assessing how variable the number of arrivals can be, providing insights into the consistency or volatility of events.

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain number. In Poisson distribution, it is computed by summing up probabilities from the smallest possible value to the target value. For instance, \( P(X \leq 10) \) signifies the cumulative probability that "at most 10" aircraft arrive.
To find "at least" probabilities like \( P(X \geq k) \), subtract the cumulative probability up to \( (k-1) \) from 1; for example, \( P(X \geq 10) = 1 - P(X < 10) \).
In our exercise, calculating \( P(X \leq 10) \) and \( P(X \geq 20) \) across different periods (using appropriate \( \mu \)) involves applying these cumulative principles. It provides a broader view of possible outcomes, essential for predictions and risk assessments.