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The Probability Mass Function (PMF) is a crucial element when dealing with discrete probability distributions, such as the Poisson distribution. It provides the likelihood of a specific number of events happening in a fixed interval. For the Poisson distribution, the PMF formula is given by:\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]

• \( P(X = k) \) is the probability of observing exactly \( k \) events (e.g., breakdowns).
• \( \lambda \) is the average rate of occurrence, here set at 1.5 breakdowns per day.
• \( e \) stands for the base of the natural logarithm, approximately equal to 2.718.
• \( k! \) denotes the factorial of \( k \), which is the product of all positive integers up to \( k \).

This function allows us to calculate the probability of 0, 1, 2, etc., breakdowns occurring. It is important to understand that each value of \( k \) we assess with the PMF provides a separate probability of that specific number of breakdowns happening within the day.

Breakdown Analysis in the context of this problem involves evaluating the likelihood of different numbers of breakdowns per day, particularly focusing on the chances of three or more breakdowns.
To begin with, we compute the probabilities using the Poisson PMF for 0, 1, and 2 breakdowns:

• Probability of 0 breakdowns: \( P(X = 0) = e^{-1.5} \)
• Probability of 1 breakdown: \( P(X = 1) = 1.5 \times e^{-1.5} \)
• Probability of 2 breakdowns: \( P(X = 2) = 1.125 \times e^{-1.5} \)

Once these probabilities are calculated, they are summed to find the total probability for fewer than three breakdowns.
This sum is then subtracted from 1 to give the probability of having three or more breakdowns on a day, which is:\[ P(X \geq 3) = 1 - ( P(X = 0) + P(X = 1) + P(X = 2) ) \]By subtracting the cumulative probability of fewer than three breakdowns from 1, we find the complement, revealing the desired probability.
This analysis is critical as it helps management understand and plan for days when the production line might experience significant disruptions.

Statistical Inference involves drawing conclusions about a population based on sample data. In our scenario with the Poisson distribution, we're inferring the likelihood of breakdowns based on parameters that describe the average number of these events.
Through the use of the Poisson probability mass function, we infer the probability of experiencing three or more breakdowns per day, which is crucial for decision-making and risk management in production processes.
The inference here is derived by:

• Defining the random variable, namely the number of breakdowns in a day.
• Establishing the average rate of breakdowns (\( \lambda = 1.5 \)).
• Calculating the probabilities for different breakdown counts and their complements.

Using these calculations, the statistical inference made is that there is approximately a 19.1% chance that a day will have three or more breakdowns.
This actionable insight can guide management in allocating additional resources or implementing preventative measures, showcasing the real-world applicability of statistical inference in operational settings.