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A discrete random variable is a type of random variable that can take on countable values. This differs from continuous random variables, which can take on any value within a range. Discrete random variables are typically involved in statistical problems related to probabilities of specific outcomes, such as rolling a die or flipping a coin.
In the context of the exercise, the variable \(Y\) is a discrete random variable. This means it can only assume values from a specified finite or countable set. For example, if \(Y\) represents the number of times a coin lands on heads in three flips, it could take the values 0, 1, 2, or 3.
Understanding discrete random variables is key to calculating both expected values and variances, as these concepts rely on the probabilities associated with each of the possible outcomes.

The expected value is a fundamental concept in probability and statistics. It represents the mean or average value that a random variable is expected to take on. For a discrete random variable, the expected value is calculated by summing up the products of each possible value of the variable and its associated probability.
For the random variable \(Y\), the expected value \(E(Y) = \mu\) can be thought of as the weighted average of all possible values of \(Y\). This value helps in understanding the typical behavior of the variable. In the exercise, the expected value of the scaled variable \(W = 2Y\), was shown to be \(E(W) = 2\mu\). This doubles the expected value since each instance of \(Y\) is doubled in \(W\).
Expected values are crucial for decision making when dealing with uncertainties, as they give the long-term average outcome of an experiment.

Variance is a statistical measure that captures the extent to which the values of a random variable differ from their expected value (mean). It quantifies the spread or dispersion of the random variable's outcomes. A higher variance indicates a wider spread of values around the mean.
The variance of a discrete random variable \(Y\), denoted \(V(Y) = \sigma^2\), is calculated by finding the expected value of the squared deviations from the mean \(\mu\). For the variable \(W = 2Y\), the variance is shown to be \(V(W) = 4\sigma^2\), i.e., four times the variance of \(Y\).
This demonstrates an important property of variance: when a random variable is scaled by a constant, its variance is scaled by the square of that constant. Thus, scaling affects variance in a more pronounced way than it affects the expected value.

A scaled random variable is created when a constant is multiplied by a random variable. In statistical contexts, scaling usually involves operations like multiplying each outcome of a random variable by the same constant.
When \(Y\) is scaled by a factor of 2 to become \(W = 2Y\), each possible outcome of \(Y\) is doubled. This has clear effects on both the expected value and the variance of the variable. As seen, the expected value of the scaled variable \(W\) is double the expected value of \(Y\), i.e., \(E(W) = 2\mu\). Similarly, the variance of \(W\) increases by a factor of four, as \(V(W) = 4\sigma^2\).
Scaling changes the data's distribution characteristics, which is why understanding these transformations helps in making predictions and understanding how changes affect the data.