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Skewness is a measure of the asymmetry in a probability distribution. When we talk about skewness, we often refer to how the tails of a distribution are behaving. If the tail on the right side of the distribution is longer or fatter, it’s called "positive skewness." Conversely, if the tail on the left is longer or fatter, it's known as "negative skewness." This concept helps us understand where the majority of data points lie and how much the distribution deviates from a symmetrical bell curve.
In the context of the exercise, changing skewness relates to flipping the distribution. By reversing the distribution of scores a student might earn on a quiz, skewness changes from positive to negative or vice versa. This happens because the contributions to skewness are mirrored, effectively flipping the distribution and changing the sign.

A random variable is a variable whose possible values represent outcomes of a random phenomenon. Random variables can be discrete, representing countable outcomes like the roll of a dice, or continuous, representing data that can take any value within a range.
In the given exercise, the random variable is denoted as \(X\), representing the score a student earns on a quiz. The possible values for \(X\) range from 0 to 10, indicating this is a discrete random variable.
To understand the problem, we redefine it as another random variable \(Y = 10 - X\). This new variable \(Y\) effectively reverses the original distribution, helping us calculate various statistical measures such as mean and skewness.

The Probability Mass Function (PMF) provides the probability that a discrete random variable is exactly equal to a certain value. It's a fundamental concept in probability, especially useful for discrete random variables.
In our exercise, the PMF of the random variable \(X\) is denoted by \(p(x)\). It gives the probabilities of each score a student can achieve on a quiz. When we define the reversed distribution through \(Y = 10 - X\), the PMF alters to \(p_Y(y) = p_X(10 - y)\).
This shows us how reversing the scores also reverses their probabilities, making it essential for comparing skewness between \(X\) and \(Y\). Understanding PMF allows us to calculate further statistical measurements needed to understand the behavior of the distribution.

Variance is an important statistical measure showing how data points in a dataset are spread out. It tells us how much the values differ from the mean and from each other.
In terms of formulas, variance is expressed as \( \sigma^2 \) and calculated by finding the average of the squared deviations from the mean.
The exercise makes sure to highlight that variance remains unchanged when reversing a distribution via a symmetrical transformation. When we move from \(X\) to \(Y = 10 - X\), the variance \( \sigma_Y^2 \) remains equal to \( \sigma_X^2 \). This consistency is crucial to understanding that while position-based measures such as the mean and skewness may alter, variance stays constant under certain symmetrical data transformations.