Notation glossary
The symbols and conventions this course commits to
Probability has more than one popular way to write almost everything. One book writes the complement of an event as \(A^{c}\), another writes \(A'\), a third writes \(\bar A\); one author calls a distribution “Exponential with rate \(\lambda\)” while the next calls it “Exponential with mean \(\lambda\)” — and the two mean opposite things. None of these choices is wrong, but a course that drifts between them quietly becomes harder to read than the mathematics deserves. So this page fixes one choice for each symbol and keeps it for the whole semester.
This glossary is the public mirror of the instructor’s private notation ledger for STAT 35003. The ledger is a build-internal file; what you read here is its student-facing version, written in the course’s own words. It is not reproduced from either course text — it simply states, in one place, the conventions the weekly notes already use. When a week note introduces a symbol, you can come here to see exactly what it commits to and why that particular form was chosen over the alternatives.
The notation choices below are stable across the whole semester; the numeric instances used to illustrate them are synthetic teaching examples (seed 35003 set), consistent with the recurring case used throughout the weekly notes.
How to read this page
The page has three parts. Core symbols is the master table — every piece of notation, its meaning, and the convention chosen when a symbol has competitors. The parameterizations we fix pins down what each named distribution’s parameters actually mean (where first-course probability most often goes silently wrong) and ties each one to the recurring “commuter’s morning” numbers. Words we keep distinct addresses the word-pairs that are routinely blurred — independent versus mutually exclusive, a density versus a probability — where the confusion is conceptual rather than typographic.
Core symbols
Every symbol below appears in the weekly notes with exactly this meaning. Where a symbol has common alternatives, the “Convention chosen” column says which form this course uses and excludes the others.
| Symbol | Meaning | Convention chosen |
|---|---|---|
| \(\Omega,\ \omega\) | the sample space and a single outcome in it | outcomes \(\omega\) are elements of \(\Omega\) |
| \(A,\ B,\ E\) | events (subsets of the sample space) | capital letters for events |
| \(P(A)\) | the probability of event \(A\) | written \(P(\cdot)\), never \(\Pr\) |
| \(A^{c}\) | the complement of \(A\) — everything in \(\Omega\) that is not in \(A\) | \(A^{c}\) only (not \(A'\), not \(\bar A\)) |
| \(A\cup B\) | the union — \(A\) or \(B\) (or both) | “or” is inclusive |
| \(A\cap B\) | the intersection — \(A\) and \(B\) together | both events occur |
| \(P(A\mid B)\) | the conditional probability of \(A\) given \(B\) | defined only when \(P(B)>0\) |
| \(A\perp B\) | \(A\) is independent of \(B\) | independence, not mutual exclusivity |
| \(X,\ Y\) | random variables | capitals for the variable, lowercase \(x,y\) for values |
| \(p(x),\ p_X(x)\) | a probability mass function (pmf) | \(p(x)=P(X=x)\), for a discrete \(X\) |
| \(f(x),\ f_X(x)\) | a probability density function (pdf) | a density, not a probability |
| \(F(x)\) | a cumulative distribution function (cdf) | \(F(x)=P(X\le x)\) |
| \(E[X]\) | the expectation (mean) of \(X\) | named \(\mu\) when we refer to the value |
| \(\operatorname{Var}(X),\ \sigma^2\) | the variance of \(X\) | same quantity, two notations |
| \(\sigma\) | the standard deviation | \(\sigma=\sqrt{\operatorname{Var}(X)}\) |
| \(\operatorname{Cov}(X,Y),\ \rho\) | covariance and correlation | correlation \(\rho\in[-1,1]\) always |
| \(X\sim\text{Dist}(\cdot)\) | “\(X\) is distributed as” | parameter forms fixed in the next section |
| \(\lambda\) | a rate | Poisson count rate / Exponential rate |
| \(\mu,\ \sigma\) | the Normal parameters | mean and standard deviation (R’s rnorm) |
A few of these deserve a sentence beyond the table.
The choice of \(P(\cdot)\) rather than \(\Pr\) is purely for a clean line; the two are identical in meaning. The complement choice — \(A^{c}\) throughout — matters more than it looks: \(\bar A\) collides with the overbar statistics later uses for a sample mean \(\bar X\), and \(A'\) collides with the prime calculus uses for a derivative. Reserving one symbol per idea keeps later weeks readable.
The conditional bar carries a precondition that is easy to forget: \(P(A\mid B)\) is only defined when \(P(B)>0\), because the definition divides by \(P(B)\) — you cannot condition on something that cannot happen.
The pmf and the cdf use deliberately different letters from the density. For a discrete variable, \(p(x)=P(X=x)\) is a genuine probability — the chance that \(X\) lands exactly on \(x\). The density \(f(x)\) is not a probability; it is a height, and only an area under it is a probability. The cdf \(F(x)=P(X\le x)\) is always a probability, discrete or continuous. Keeping \(p\), \(f\), and \(F\) visually distinct reminds you which object you are holding.
The parameterizations we fix
Naming a distribution is not enough. “Exponential,” “Normal,” and “Geometric” each have two equally popular parameterizations in circulation, and choosing the wrong one flips a mean into a rate or a standard deviation into a variance. This section states the single form the course uses for each model and anchors it to the recurring commuter scenario.
Bernoulli and Binomial. A Binomial counts successes in a fixed number of independent trials, each with the same success probability \(p\):
\[ X \sim \text{Binomial}(n,p), \qquad E[X] = np, \qquad \operatorname{Var}(X) = np(1-p). \]
The recurring instance is the quiz-guessing thread: a ten-question true/false quiz answered by pure guessing is \(X\sim\text{Binomial}(10,\,0.5)\), so its mean is \(np = 10(0.5) = 5\) correct and its variance is \(np(1-p) = 10(0.5)(0.5) = 2.5\), giving a standard deviation of about \(1.58\). (These hold the course’s locked numbers; data are synthetic, seed 35003 set.)
Geometric. This is the parameterization most often gotten “off by one.” The course counts the number of trials up to and including the first success, so the support starts at \(1\):
\[ X \sim \text{Geometric}(p), \qquad x \in \{1, 2, 3, \dots\}, \qquad E[X] = \frac{1}{p}. \]
Some texts and some software instead count the number of failures before the first success, which starts the support at \(0\) and changes the mean. The course always uses the trials version (support \(\{1,2,\dots\}\)); the labs flag where R’s built-in *geom functions use the failures version instead, so you are never surprised by the shift.
Poisson. A Poisson models a count of events in a fixed window, with a single rate parameter \(\lambda\) that is simultaneously the mean and the variance:
\[ N \sim \text{Poisson}(\lambda), \qquad E[N] = \operatorname{Var}(N) = \lambda. \]
The recurring instance is shuttle arrivals: shuttles arrive at rate \(\lambda = 4\) per hour, so the number of arrivals in an hour is \(N\sim\text{Poisson}(4)\), with both mean and variance equal to \(4\).
Uniform. The continuous Uniform spreads probability evenly across an interval:
\[ X \sim \text{Uniform}(a,b), \qquad f(x) = \frac{1}{b-a} \ \text{ for } a \le x \le b. \]
Unless a note explicitly says “discrete uniform,” “Uniform” means this continuous version on \([a,b]\).
Exponential. This is the parameterization that most often reverses its meaning between sources. The course uses the rate form, where the parameter \(\lambda\) is a rate and the mean is its reciprocal:
\[ T \sim \text{Exponential}(\lambda), \qquad f(t) = \lambda e^{-\lambda t} \ \text{ for } t \ge 0, \qquad E[T] = \frac{1}{\lambda}. \]
The recurring instance is the wait for the next shuttle. With shuttles arriving at rate \(\lambda = 4\) per hour, the wait \(T\sim\text{Exponential}(4/\text{hr})\) has mean \(1/\lambda = \tfrac14\) hour, which is 15 minutes. A mean-parameterized “Exponential(15 min)” would describe the same physical situation but would read the number differently — which is exactly why the course pins the rate form and states it each time.
Normal. The course parameterizes the Normal by its mean and standard deviation, matching R’s rnorm(n, mean, sd), not by mean and variance:
\[ X \sim \text{Normal}(\mu, \sigma), \qquad f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\,(x-\mu)^2 / (2\sigma^2)}. \]
The recurring instance is the morning commute time \(C\sim\text{Normal}(\mu = 22\text{ min},\, \sigma = 5\text{ min})\) — that is a mean of \(22\) minutes and a standard deviation of \(5\) minutes, not a variance of \(5\). Whenever a note writes \(N(22,5)\), the second number is the standard deviation.
Words we keep distinct
Some confusions are not about which symbol to write but about which idea you actually mean. These five word-pairs are the ones a first probability course most often blurs, and the weekly notes keep them strictly apart.
Probability versus conditional probability. \(P(A)\) is the chance of \(A\) with no information assumed; \(P(A\mid B)\) is the chance of \(A\) once you know \(B\) has occurred. They are usually different numbers. In the shuttle thread, the unconditional chance the shuttle is on time is \(P(\text{on time}) = 0.81\), but once you know it is raining the chance drops to \(P(\text{on time}\mid\text{rain}) = 0.60\). The conditioning bar changes the question, so it changes the answer.
Independent versus mutually exclusive. These are not the same property, and they are not even close. Two events are independent (\(A\perp B\)) when knowing one happened tells you nothing about the other — formally, \(P(A\cap B) = P(A)\,P(B)\). Two events are mutually exclusive when they cannot both happen, so \(P(A\cap B) = 0\). Two mutually exclusive events with positive probability are about as far from independent as possible, since learning one occurred tells you the other definitely did not. In the shuttle thread, “on time” and “rain” are not independent: \(P(\text{on time}\mid\text{rain}) = 0.60\) differs from \(P(\text{on time}) = 0.81\), so rain carries information about being on time. They are also not mutually exclusive — the shuttle can be on time on a rainy day. The two ideas are simply different axes.
A pmf versus a density. For a discrete variable, \(p(x) = P(X=x)\) is an honest probability: it is the chance \(X\) equals exactly \(x\), and these values sum to \(1\). For a continuous variable, the density \(f(x)\) is not a probability — the chance of any single exact value is \(0\). The density is a height; probability is the area under that height across an interval, \(P(a\le X\le b) = \int_a^b f(x)\,dx\). A density can even exceed \(1\) at a point (a tall, narrow peak) without anything being wrong, because no single height is a probability. Reading \(f(x)\) as if it were \(P(X=x)\) is the classic continuous-variable error, and the distinct letters \(p\) and \(f\) exist precisely to head it off.
Variance versus standard deviation. These measure the same spread on two different scales. The variance \(\operatorname{Var}(X) = \sigma^2\) is in squared units; the standard deviation \(\sigma = \sqrt{\operatorname{Var}(X)}\) is back on the variable’s own scale. For the quiz thread, \(\operatorname{Var}(X) = 2.5\) (in “squared questions,” which is not a natural unit), while the standard deviation \(\sigma = \sqrt{2.5}\approx 1.58\) is in questions, the same units as \(X\) itself. When you want a number you can compare to the variable directly, you want \(\sigma\).
Covariance versus correlation. Both describe how two variables move together, but on very different scales. The covariance \(\operatorname{Cov}(X,Y) = E[XY] - E[X]\,E[Y]\) is unbounded — its size depends on the units and spread of \(X\) and \(Y\), so its magnitude alone tells you little. The correlation rescales it, \(\rho = \operatorname{Cov}(X,Y)\,/\,(\sigma_X\,\sigma_Y)\), and is always trapped in \([-1, 1]\). In the joint rain-and-lateness thread, the covariance is \(\operatorname{Cov}(X,Y) = 0.063\) — a small number hard to interpret on its own — while the correlation \(\rho \approx 0.35\) reads at a glance as a moderate positive relationship: rain and lateness tend to go together. The sign is the same in both; the interpretability is what the correlation buys you.
How this connects to the rest of the course
This glossary is a reference, not a lesson — the ideas behind these symbols are developed, with full worked examples, in the weekly notes. The complement and the inclusive “or” arrive in Week 2 — sample spaces, events & rules; the conditional bar and its \(P(B)>0\) precondition in Week 3 — conditional probability; and the independence symbol \(A\perp B\), with the contrast against mutual exclusivity, in Week 4 — independence & information. Expectation, variance, and standard deviation are built in Week 8 — expectation & variance; covariance and correlation in Week 12 — joint distributions & dependence.
For the parameter forms, the companion distribution reference lays out each model’s pmf or density, mean, and variance side by side, and the R & Quarto setup page shows how R’s argument order lines up (or, for the Geometric, deliberately does not) with the conventions here. You can also return to the resources index or the syllabus.
Public vs. graded
These notes, the examples, and the practice here are public and ungraded — study material only. No graded prompts, answer keys, rubrics, point values, or due dates appear on this site. Graded checkpoints, quizzes, homework, labs, the midterm, the project, and the final live in Blackboard (the LMS), which is authoritative for due dates, submissions, and grades. If this page and Blackboard ever disagree, follow Blackboard.