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The probability mass function (PMF) is a fundamental concept in statistics, particularly when dealing with discrete probability distributions. It describes the probability of a discrete random variable taking on specific values.

For the Poisson distribution, the PMF is defined as follows:
\( P(X = k) = \frac{e^{-\text{μ}} \text{μ}^k}{k!} \)
Here:

• \( X \) is the random variable representing the number of occurrences
• \( μ \) is the average rate (or mean) of occurrence
• \( k \) is the number of occurrences we are interested in
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

In our example, we want to find the probability that \( X = 6 \) when \( μ = 5.2 \). By substituting these values into the PMF formula, we get:
\[ P(X = 6) = \frac{e^{-5.2} \times 5.2^6}{6!} \]
This is the key to solving Poisson distribution problems, as it allows us to compute exact probabilities for specific occurrences.

The average rate, often denoted as \( μ \), is a crucial parameter in the Poisson distribution. It represents the mean number of occurrences in a given interval.

Consider the Poisson distribution as a model for counting occurrences, such as:

• The number of emails received in an hour
• The number of customer arrivals in a store
• The number of defects in a batch of products

In our example, the average rate \( μ \) is specified as 5.2. This means, on average, 5.2 events happen per given interval period.

Understanding the average rate helps in setting up the problem, as it directly influences the probability distribution of possible outcomes. In the PMF formula:
\[ P(X = k) = \frac{e^{-\text{μ}} \text{μ}^k}{k!} \]
the parameter \( μ \) appears both in the exponent in the term \( e^{-\text{μ}} \), and as the base of the power term \( \text{μ}^k \). The average rate thus directly impacts the shape and probabilities of the Poisson distribution.

Factorials play an essential role in the Poisson distribution's probability mass function. A factorial, denoted as \( k! \), represents the product of all positive integers up to \( k \). For example:

• \(2! = 2 \times 1 = 2 \)
• \(3! = 3 \times 2 \times 1 = 6 \)
• \(4! = 4 \times 3 \times 2 \times 1 = 24 \)

In our Poisson distribution formula:
\[ P(X = k) = \frac{e^{-\text{μ}} \text{μ}^k}{k!} \]
the factorial function \( k! \) appears in the denominator. It scales the probabilities by accounting for the number of ways \( k \) events can occur.

In our specific case where \( k = 6 \), we need to compute \( 6! \):
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
Substituting this into our formula along with the values for \( e^{-5.2} \) and \( 5.2^6 \), we can accurately compute the desired probability. Understanding factorials helps in simplifying and solving such probability problems.