A probability mass function, or PMF, is the rule that assigns a probability to each possible value of a discrete random variable. If a random variable counts something — defective parts in a batch, cars arriving at an intersection, packets dropped in a network — then its PMF is the complete description of how likely each count is. Written as p(x) = P(X = x), it is the discrete counterpart to the probability density function used for continuous variables.
Two properties make a function a legitimate PMF: every probability must be non-negative (0 is less than or equal to p(x), which is less than or equal to 1), and the probabilities over all possible values must sum to exactly 1. These two conditions are the test you apply whenever you are handed a table or formula and asked whether it represents a valid distribution, or when you must solve for an unknown constant that makes it valid. They also underpin everything later in the module, since expected value and variance are weighted sums that only make sense when the weights are valid probabilities.
As you watch, pay attention to three things: how a PMF is displayed (table, formula, or bar graph), how the summation property is used as an algebraic tool rather than just a definition, and how individual PMF values are interpreted in words — for example, "there is a 12% chance that exactly two units fail." Also notice the distinction between a PMF and a cumulative distribution function (CDF), which accumulates probability up to a value.
A visual walkthrough of probability distributions for random variables, covering the probability mass function, the probability density function, and the cumulative distribution function. Based on its title and description, it uses graphical representations to show what each function reports and how discrete and continuous cases differ. Expect the discrete PMF to be presented as probability assigned to individual outcomes, with total probability accounted for across the distribution.
It gives the conceptual foundation and vocabulary for the whole topic, and placing the PMF alongside the PDF and CDF helps you avoid the most common confusion in this module. The visual approach makes the non-negativity and sum-to-one properties intuitive rather than purely symbolic.
A short, focused worked example showing how to demonstrate that a given function is a valid probability mass function of a discrete random variable. The description indicates the video walks through the verification process, which typically means checking that all values are non-negative and that the probabilities sum to one, and may involve solving for a constant that makes the sum equal 1.
This is the most direct match to the validation and construction subtopics, and it models the exact style of problem you will be asked to solve. At just over six minutes it is an efficient review before attempting homework.
Steve Brunton extends the idea of a distribution to two random variables at once, introducing the joint probability distribution and how it quantifies dependence between X and Y. The description suggests coverage of what happens when variables are independent versus dependent, which connects to marginal distributions obtained by summing over one variable.
It is useful enrichment that shows the summation-to-one property generalizing to two dimensions, reinforcing that a PMF is fundamentally a complete, normalized accounting of probability. Brunton is a well-known engineering faculty lecturer whose explanations are reliable and clearly motivated.
Together these videos build directly toward MO1: the first establishes what a PMF is and how it relates to other distribution functions, and the second demonstrates the non-negativity and summation checks used to construct and validate a discrete distribution. Because expected value and variance (MO2) are computed as probability-weighted sums over a PMF, mastering the PMF here is a prerequisite for those calculations, and for recognizing the binomial and Poisson distributions later as specific PMFs with defining conditions and parameters (MO3, MO4). The emphasis on reading and stating what an individual probability value means supports MO5, where you must translate computed numbers into defensible conclusions about an engineering problem. The joint distribution video is optional depth that strengthens your general sense of how probability must be allocated across all outcomes.
These videos were chosen from YouTube on October 2, 2026. Each has captions; the player turns them on. Videos on YouTube can be changed or removed — if one no longer plays, tell your instructor. If a player does not appear, use its Watch on YouTube link.