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The probability mass function (PMF) of a Poisson distribution is a mathematical expression used to determine the probability of a certain number of events occurring within a fixed interval. In Poisson's case, the focus is on the number of occurrences in a given space or time, such as the number of plant neighbors in a gardening area. The Poisson PMF is defined as: \\[P(X=k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!}\] where:\

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• \(X\) is the random variable representing the number of events.
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• \(k\) is the number of events.
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• \(\lambda\) is the average number of events in that interval.
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• \(e\) is the base of natural logarithms, approximately equal to 2.71828.
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• \(k!\) is the factorial of \(k\), which is the product of all positive integers up to \(k\).
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\This function tells us how probabilities are distributed for differently sized event space intervals (e.g., different area sizes for plant seedlings). It's crucial for calculating probabilities in scenarios where events are rare and time or space-constrained.

Calculating the mean in a Poisson distribution is straightforward and essential for setting up the problem correctly. The mean, denoted as \(\lambda\), is calculated as follows:\\[\lambda = A \times d\] where:\

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• \(A\) represents the area size.
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• \(d\) is the density of occurrences (e.g., seedlings per unit area).
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\For the problems discussed, calculating \(\lambda\) set the stage for understanding the frequency with which the events (seedlings) occur in a given space. For instance, if \(A=1\) square meter and \(d=4\) seedlings per meter, then \(\lambda = 4\). Similarly, if \(A=2\) and \(d=4\), then \(\lambda = 8\). Knowing \(\lambda\) is crucial since it directly feeds into the PMF calculations to give precise probabilities.

When it comes to the Poisson distribution, calculating the probability of different outcomes is key to understanding expected events within a space. Let's break down calculating probabilities using examples: \
For example (a), when no neighbors exist within 1 meter, we are looking for \(P(X=0)\). Using the PMF: \\[P(X=0) = \frac{4^0 \cdot e^{-4}}{0!} = e^{-4}\] \(\approx 0.0183\), showing a low chance in this scenario. \
For example (b), for at most three neighbors within 2 meters, we sum individual probabilities: \

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• \(P(X=0) = e^{-8}\)
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• \(P(X=1) = 8e^{-8}\)
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• \(P(X=2) = 32e^{-8}\)
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• \(P(X=3) = 85.33e^{-8}\)
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\The sum gives \(P(X \leq 3) \approx 0.00034\), a small total probability, reflecting rarity in the scenario. \Understanding how to sum these probabilities informs us on how likely it is to encounter different numbers of neighbors, providing insight into density effects over varied areas.