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Probability is a fundamental concept in mathematics, crucial for determining the likelihood of various outcomes. It offers a numerical value to the likelihood of an event, typically ranging from 0 to 1. A probability of 0 indicates that an event will not occur, while a probability of 1 indicates certainty. In our exercise of rolling a fair die, each face of the die has an equal probability of appearing:

• Each number (1 through 6) has a probability of \(rac{1}{6}\).

This is because there are six faces, all equally likely to land facing up on a fair die.
When calculating the expected value of a random variable such as the square of the number on the die, understanding probabilities allows us to correctly weigh each outcome by its chance of occurrence. By combining probabilities with values, we compute important statistical properties that describe the distributions of random processes.

A random variable is a variable whose possible values are outcomes of a random phenomenon. It serves as a bridge between probabilistic concepts and real-world phenomena, often allowing us to apply mathematical rigor to situations with inherent uncertainty. In our context, the random variable \(X\) is defined as the square of the number rolled on a die. This turns a simple number into a squared outcome:

• For instance, if we roll a 2, \(X\) becomes \(2^2=4\).

Random variables can be classified into discrete and continuous. Here, \(X\) is discrete because it takes on specific, distinct values based on the outcome of the die roll. Understanding this helps us determine how to approach calculations and predictions based on the random variable. Different outcomes result in different values of \(X\), each occurring with a certain probability, which we use to calculate things like the expected value.

A discrete distribution is a probability distribution that shows the probabilities of outcomes with discrete values. These are specific and countable, like the outcomes of rolling a die. In our exercise, the values of the random variable \(X\) are distinct numbers: 1, 4, 9, 16, 25, 36. Each of these is the square of a number from the die roll.
The probabilities associated with each value are all \(\frac{1}{6}\) because the die is fair. This uniform probability across outcomes is a key feature of a discrete uniform distribution, where each outcome is just as likely as any other.

• Discrete distributions are often represented using probability mass functions, which sum to 1 over all possible outcomes.

Applying discrete distribution theory allows us to compute expected values and other useful statistics that characterize the behavior of random variables like \(X\). Understanding this framework empowers us to make informed predictions and decisions based on probabilistic data.