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The Probability Mass Function (PMF) is a crucial concept in understanding discrete probability distributions, like the Poisson distribution used in our example. It defines how the probability is distributed among different possible values a discrete random variable, such as the number of phone calls, can take.

Mathematically, the PMF for a Poisson-distributed variable is given by:\[ p(y; \lambda) = \frac{e^{-\lambda} \lambda^y}{y!} \] Here, \( \lambda \) is the average rate (mean) of occurrences and \( y \) is the specific value of interest.

This function enables us to compute the likelihood of exactly \( y \) events happening in a fixed interval. In our example, the PMF helps calculate the probability that there will be exactly 5 or 6 phone calls in a day, showing that these two counts are equally likely.

The expected value, often referred to as the mean, is a key statistical measure that indicates the average or central value of a random variable's possible outcomes.

In the context of a Poisson distribution, this is represented by \( \lambda \), which, in our exercise, is 6.

This means that, on average, we expect 6 phone calls to the fire department per day. The expected value is calculated in such a way that it reflects the center of the probability distribution, providing insights into what one might anticipate over many repetitions of the experiment.

Understanding the expected value is important as it helps in predicting outcomes and preparing for various scenarios based on probabilities. In decision-making contexts, knowing the expected value can guide resource allocation for different possible events.

A discrete probability distribution describes the probability of each possible outcome for a discrete random variable.

Discrete random variables, unlike continuous ones, can take on a finite or countably infinite set of values.

The Poisson distribution, specifically, is a type of discrete probability distribution that measures the probability of a given number of events occurring in a fixed interval of time or space.

Characteristics of a Poisson distribution include:

• Events occur independently.
• The average rate (\( \lambda \)) at which events happen is constant.
• Two events cannot occur at the exactly same instant.

In our example, the number of phone calls received by the fire department is modeled by this type of distribution, making it possible to calculate probabilities for different call volumes.

Mathematical statistics is the branch of mathematics that uses probability theory to estimate population parameters and test hypotheses based on sample data.

In our exercise, mathematical statistics are employed to derive probabilities and demonstrate conclusions about real-world events, like the frequency of phone calls.

By understanding mathematical statistics, one can not only describe data more meaningfully but also make predictions or informed decisions on an empirical basis. Using distributions like Poisson, statisticians can characterize the nature of datasets and infer trends or patterns.

In practice, mathematical statistics allows one to determine the most likely outcomes and verify these outcomes analytically, as shown by confirming that \( p(5) = p(6) \) using the probability mass function in this exercise. Through careful statistical analysis, patterns and probabilities in data can be accurately described and leveraged.