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Discrete random variables are an essential concept in statistics and probability, especially when dealing with countable outcomes. In the given exercise, the discrete random variable, denoted as \(Y\), represents the number of moving violations an individual received over the last three years. This variable can take on specific, separate values such as 0, 1, 2, or 3.
A discrete random variable contrasts with a continuous random variable, which can take on any value in a range. Understanding discrete random variables is crucial because they help us determine probabilities for different outcomes. They are often used in real-life situations, like counting the number of events happening over a fixed period. By analyzing these variables, we can make predictions and informed decisions based on likelihoods.

The probability mass function (PMF) is a powerful tool for dealing with discrete random variables. A PMF assigns probabilities to each possible value that a discrete random variable can take. In our exercise, the PMF for \(Y\) is given by the values \(p(y)\): 0.60 for \(y=0\), 0.25 for \(y=1\), 0.10 for \(y=2\), and 0.05 for \(y=3\).
A PMF is particularly useful because it provides a complete description of the probability distribution of a discrete random variable. The probabilities in a PMF always sum up to 1. This is necessary because one of the values must occur — a fundamental property of probability distributions. Using the PMF, we can calculate important statistical metrics, such as the expected value of the variable.

Calculating an expected surcharge involves applying the concept of expected value to monetary outcomes. In our exercise, the surcharge is determined using the formula \(100Y^2\), meaning the surcharge increases with the square of the number of violations.
To find the expected surcharge, we need to evaluate the expected value of \(100Y^2\). This is done by summing up the products of \(100y^2\) and the respective probabilities from the PMF. The results for \(y=0, 1, 2,\) and \(3\) are combined to give us the expected surcharge of 110 units.
Understanding the expected surcharge is important because it allows insurance companies to predict average costs associated with customer claims. This information can be critical for setting premiums and managing financial risk.

Statistics problem solving often involves identifying the right methods and models to apply to a given situation. In this exercise, we are tasked with calculating both an expected value and an expected surcharge based on a discrete random variable and its PMF.
A structured approach is key to tackling these problems efficiently:

• Identify the type and nature of the random variables involved.
• Choose the appropriate statistical model or formula, like the PMF or expected value formula.
• Substitute the given data into these formulas.
• Calculate each term methodically, checking work at each step.
• Sum the results to find the final answer.

By practicing these steps and understanding the underlying concepts, students can improve their problem-solving skills in statistics. This structured approach is applicable to a wide range of statistical and probabilistic scenarios.