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In a Poisson process, probabilities are calculated using the Poisson probability mass function. This function is used to find the likelihood of a specific number of events occurring within a fixed time period.
For example, if you want to calculate the probability of exactly 6 small aircraft arriving in a 1-hour period, where the arrival rate is 8 aircraft per hour, you use the formula:

• \( P(X = 6) = \frac{e^{-8} \cdot 8^6}{6!} \)

Here, \( e \) is the base of the natural logarithm, and "!" represents a factorial.
For cumulative probabilities, like finding the probability of at least 6 arrivals, you calculate the probability of having up to 5 arrivals and subtract it from 1:

• \( P(X \geq 6) = 1 - P(X \leq 5) \)

Cumulative distribution functions are practical tools to find probabilities over a range of events.
Such calculations help in understanding and predicting event patterns efficiently.

The expected value in a Poisson process is directly tied to its parameter \( \lambda \). This parameter represents the average number of events expected in a given time period. In the context of our exercise, \( \lambda \) signifies the mean number of aircraft arrivals, which in a 1.5-hour duration is calculated by multiplying the arrival rate per hour (8) by the time in hours (1.5):

• \( \lambda = 8 \times 1.5 = 12 \)

Thus, the expected value, or \( E(X) \), during this 90-minute period is 12.
The expected value is crucial because it gives a central measure of the distribution, reflecting where most data points are likely to cluster.
Understanding this concept helps in forming insights into future events and aids in decision-making processes.

The standard deviation in a Poisson process offers insight into the variability of the number of events around the expected value. For a Poisson distribution, the standard deviation is the square root of its expected value, \( \lambda \).
Using our example of a 90-minute time period where the expected value \( \lambda \) is 12, the standard deviation \( \sigma \) can be calculated as:

• \( \sigma = \sqrt{12} \)

This value represents the typical deviation from the mean number of arrivals, showing how spread out the arrivals could be around 12.
By understanding the standard deviation, one can gauge how predictable or variable a Poisson process might be. Higher deviations indicate more variability, while smaller deviations indicate arrivals closer to the expected number of events.