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A probability distribution describes how likely different outcomes are in a random process. For our case, we're looking at the Poisson distribution, which is a common model used for count-based data. It helps us understand the probabilities of a given number of events, like airplane arrivals, happening within a fixed interval of time or space. In mathematical terms, the Poisson distribution gives us the probability that a certain number of events, denoted as \( x \), will happen, given an average rate \( \lambda \). This formula is helpful when we know the average events but want to predict specific occurrences.

To construct a probability distribution for airplane arrivals, we take this average rate (\( \lambda = 1.4 \) arrivals per hour) and apply the Poisson formula for various \( x \) values, like 0, 1, 2, etc. This approach helps to visualize the likelihood of different numbers of arrivals, making it easier to predict and prepare for such events.

The expected number in the context of a Poisson distribution is a key concept. It tells us the average or mean number of events we anticipate over a certain period. Here, the expected number of private airplane arrivals per hour is given as \( \lambda = 1.4 \). This number represents our average rate of occurrences over the specified period.

This average is critical because it allows us to set our expectations realistically. Whether planning resources or preparing for operational tasks, knowing this average helps. It's a statistical cornerstone that feeds into the calculations of specific probabilities for individual event counts. Knowing the expected number sets the base for further statistical analysis.

Calculating probabilities in a Poisson distribution is straightforward once you grasp the basics. The formula is:

\[ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \]

Here's what each term means:

• \( \lambda \): the average rate (1.4 in our scenario).
• \( k \): the specific number of events (e.g., 0, 1, 2).
• \( e \): Euler's number, approximately 2.71828.
• \( k! \): factorial of \( k \), which is the product of all positive integers up to \( k \).

For instance, to find the probability of no airplanes arriving (\( k = 0 \)), substitute these values into the formula. With \( \lambda = 1.4 \), it simplifies to:

\[ P(X = 0) = \frac{1.4^0 \cdot e^{-1.4}}{0!} = e^{-1.4} \]

Understanding this process is crucial for interpreting statistical data and making informed predictions.

Once probabilities for different values of \( x \) are calculated, they can be organized into a probability table. This table acts as a summary of potential outcomes and their associated probabilities.

Start by calculating the probability for each \( x \) value: 0, 1, 2, and so on until reaching a logical stopping point like \( x = 10 \). Fill these values into the table:

• For \( x = 0 \), use the calculation \( e^{-1.4} \).
• For \( x = 1 \), apply the formula \( \frac{1.4^1 \cdot e^{-1.4}}{1!} \).
• Continue this process for each \( x \).

The table aids in quick reference and comparison, showcasing how likely different numbers of arrivals are within a given time. It's an efficient way to convey a dense set of potential outcomes and probabilities, facilitating better decision-making and planning.