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The Probability Mass Function (PMF) is a fundamental concept in probability theory and statistics, especially when dealing with discrete random variables like in a Poisson distribution. For a Poisson distribution, the PMF describes the probability of observing a certain number of events, \( x \), in a fixed interval. This is expressed mathematically by: \[ P(X = x) = \frac{e^{-\mu} \mu^x}{x!} \] where \( e \) is the Euler's number (approximately 2.71828), \( \mu \) is the average number of events, and \( x! \) denotes the factorial of \( x \).In our context, with \( \mu = 4 \), if we're calculating for \( x = 2 \), we plug into the formula to find the exact probability of exactly 2 events occurring. The PMF helps to solve such problems by providing a systematic way to calculate these probabilities plainly and accurately.

Calculating probabilities using the Poisson distribution involves using the PMF and involves a few straightforward steps. For example, to find \( P(X = 2) \) where \( \mu = 4 \), you substitute into the PMF as follows:

• Compute \( e^{-4} \), which is roughly 0.0183156.
• Calculate \( 4^2 \), which results in 16.
• Factorial of 2 (2!), which equates to 2.

Then, substitute these values into the PMF:\[ P(X = 2) = \frac{0.0183156 \cdot 16}{2} = 0.146525 \]This result shows that there is approximately a 14.65% chance of precisely 2 events occurring in this Poisson framework. This method is repeated similarly for other values of \( x \), making Poisson calculations methodical and efficient.

The complement rule is a helpful tool in probability, particularly when calculating the probability of a statement that can easily be derived from the complement of a simpler statement. In the Poisson distribution problem, the complement rule helps in finding \( P(X > 2) \) by using \( P(X \leq 2) \). Given that the total probability must sum to 1, the probability of \( X > 2 \) is calculated as:\[ P(X > 2) = 1 - P(X \leq 2) \]Using the data, if \( P(X \leq 2) = 0.238103 \), then:\[ P(X > 2) = 1 - 0.238103 = 0.761897 \]This implies a 76.19% chance of more than 2 events occurring. The complement rule simplifies computations by requiring you to calculate only the simpler probability and subtracting it from one to find its complement, the desired probability.