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Discrete Random Variables are fundamental concepts in probability and statistics. They represent variables that can take on a countable number of distinct values. In the context of the building permit example, the variable \( Y \) can be 1, 2, 3, 4, or 5, each representing the number of forms required.
Each possible value of a discrete random variable is associated with a specific probability. Unlike continuous random variables, which can take on infinite values within a range, discrete random variables have distinct, separate values.
They are often used in scenarios where outcomes are counted, such as the number of dice rolls, the number of students present, or, as in this case, the number of forms.

The Probability Mass Function (PMF) is a key tool for describing discrete random variables. It assigns probabilities to each possible value of a discrete random variable. In our example, the PMF is described as \( p(y) = ky \), where \( k \) is a constant that ensures probabilities sum to 1.
A PMF communicates the likelihood of each outcome clearly and quantitatively. For each value \( y \) the PMF gives us \( p(y) \), the probability that the random variable \( Y \) equals \( y \).

• The PMF must satisfy two conditions: all probabilities \( p(y) \) must be non-negative, and the sum of \( p(y) \) over all possible values of \( Y \) must equal 1.

This guarantees a valid probability distribution where all outcomes are considered, ensuring that there is no missing probability.

The sum of probabilities is a critical concept when working with probability distributions. For discrete random variables, the total probability must add up to 1. This reflects the certainty that one of the possible outcomes must occur.
In our exercise, we used the fact that \( \sum_{y=1}^{5} p(y) = 1 \) to find the constant \( k \). This equation ensures that all probabilities are accounted for, balancing the distribution effectively.

• Each individual probability \( p(y) \) is a fraction of the whole. The proportions are managed so that they collectively complete to 1.
• This also helps to verify the correctness of a PMF—for example, checking if a proposed function like \( p(y) = \frac{y^2}{50} \) can be a PMF by checking if their sum equals 1.

Proportional Probability refers to the case where probabilities are directly dependent on some variable in a proportional fashion. In other words, the probability increases or decreases in proportion to the variable's value.
In the given exercise, \( p(y) = ky \) describes a scenario where the probability of requiring \( y \) forms grows with \( y \)—meaning that needing more forms is more probable. This proportionality simplifies calculations by clearly outlining how one variable affects probability.

• To find \( k \), the sum of all proportional probabilities is set to 1, ensuring the probabilities form a valid PMF.
• Understanding such a distribution requires understanding how the weights (y-values in this case) influence probabilities, maintaining a coherent and predictable structure.

These concepts highlight the powerful, structured way probabilities can be assigned and understood, especially when they exhibit proportional traits.