Glossary and distribution reference
This is the volume's reference card, not another lesson: nothing here is derived. The first table takes every idea the volume introduces and names the part that builds it. The second is the distribution table — pmf or pdf, mean, variance, moment-generating function and conjugate prior for each family — so a symbol you meet later resolves in one look. Type in the box to filter both tables at once.
How to use this card
Filter, then follow the link
Every row of the first table links its idea to the part that introduces it, in the order the volume teaches them. The second table collects the named distributions; the conjugate-prior column is the bridge to Bayesian inference in the companion volume, where those pairs are the reason closed-form posteriors exist at all. Nothing here restates a proof — use it to find the part, then read the part.
No row matches that — try a looser word, or clear the box.
Terms, in the order they appear
From the part that builds it
| Term | Meaning | Introduced in |
|---|---|---|
| Sample space, outcome, event | $\Omega$ is the set of all outcomes; an event is a subset of it. | Part 1 |
| Axioms of probability | Non-negativity, normalisation, countable additivity. | Part 1 |
| Relative frequency | S_n/n, the proportion of successes in n trials. | Part 1 |
| Odds, log-odds, coherence | p/(1-p) and its log; incoherent beliefs admit a Dutch book. | Part 1 |
| Permutation, combination, binomial coefficient | n!/(n-k)!, $\binom{n}{k}=n!/(k!(n-k)!)$. | Part 2 |
| Birthday problem, collision probability | Chance of a repeated value when drawing from a finite set. | Part 2 |
| Conditional probability | $P(A\mid B)=P(A\cap B)/P(B)$. | Part 3 |
| Multiplication / chain rule | $P(A\cap B)=P(B)P(A\mid B)$ and its extension. | Part 3 |
| Law of total probability | Summing over a partition of the sample space. | Part 3 |
| Bayes' rule | $P(D\mid +)\propto P(+\mid D)P(D)$; posterior, prior, likelihood. | Part 4 |
| Base-rate fallacy, likelihood ratio | The prior dominates a rare-event test; LR multiplies the odds. | Part 4 |
| Independence, conditional independence | $P(A\cap B)=P(A)P(B)$; the conditional version can hold when the plain one fails. | Part 5 |
| Explaining-away, Simpson's paradox | A collider induces dependence; pooling can reverse an association. | Part 5 |
| Random variable, pmf, cdf | A function from $\Omega$ to $\mathbb{R}$; $F(x)=P(X\le x)$. | Part 6 |
| Expectation | $\mathbb{E}[X]=\sum_k k\,P(X=k)$; linear whatever the dependence. | Part 7 |
| Variance, standard deviation | $\operatorname{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X]^2$. | Part 8 |
| Markov, Chebyshev, Hoeffding | Tail bounds that need only moments or a bounded range. | Part 8 |
| Discrete families | Bernoulli, binomial, geometric, negative binomial, Poisson. | Part 9 |
| Poisson limit | Binomial with $np\to\lambda$ converges to Poisson. | Part 9 |
| Probability density | $P(a\le X\le b)=\int_a^b f(x)\,dx$; f is not a probability. | Part 10 |
| Continuous families | Uniform, exponential, gamma, beta, Student-t, Gaussian. | Part 11 |
| Max-entropy characterisation | The Gaussian is the maximum-entropy density for a given variance. | Part 11 |
| Change of variables, inverse-CDF sampling | Densities rescale by |g'|; F^{-1}(U) samples any distribution. | Part 12 |
| Joint, marginal, conditional | p(x,y), its row and column sums, and $p(y\mid x)$. | Part 13 |
| Covariance, correlation | $\operatorname{Cov}(X,Y)$; $\rho$ is the normalised version. | Part 14 |
| Covariance matrix $\Sigma$ | Symmetric positive semi-definite; its eigenvectors are the ellipse axes. | Part 14 |
| Multivariate Gaussian | $\mathcal{N}(\mu,\Sigma)$; conditioning, marginals, Mahalanobis distance. | Part 15 |
| Whitening | $\Sigma^{-1/2}(x-\mu)$ makes the components independent standard normals. | Part 15 |
| Convolution, MGF | The density of a sum is a convolution; mgfs turn it into a product. | Part 16 |
| Law of large numbers | $\bar X_n\to\mu$ in probability and almost surely. | Part 17 |
| Modes of convergence | In probability, almost sure, in mean square, in distribution. | Part 17 |
| Central limit theorem | $\sqrt{n}(\bar X_n-\mu)/\sigma\Rightarrow\mathcal{N}(0,1)$. | Part 18 |
| Berry–Esseen bound | The $O(1/\sqrt n)$ rate of CLT convergence. | Part 18 |
| Heavy tails, concentration of measure | Where the CLT fails; why high dimensions are counter-intuitive. | Part 19 |
| σ-algebra, measurable set | The events a probability is allowed to assign; Vitali sets are not measurable. | Part 20 |
| Measure, filtration, Borel–Cantelli | A probability is a measure with total mass one; filtrations encode information over time. | Part 20 |
The distribution reference card
pmf or pdf, mean, variance, mgf, conjugate prior
| Family | pmf / pdf | Mean | Variance | MGF | Conjugate prior for |
|---|---|---|---|---|---|
| Bernoulli (p) | p^k(1-p)^{1-k} | p | p(1-p) | 1-p+pe^t | Beta (itself) |
| Binomial (n,p) | $\binom{n}{k}p^k(1-p)^{n-k}$ | np | np(1-p) | (1-p+pe^t)^n | Beta |
| Geometric (p), on $1,2,\dots$ | (1-p)^{k-1}p | 1/p | (1-p)/p^2 | $\dfrac{pe^t}{1-(1-p)e^t}$ | Beta |
| Negative binomial (r,p) | $\binom{k-1}{r-1}p^r(1-p)^{k-r}$ | r/p | r(1-p)/p^2 | $\left(\dfrac{p}{1-(1-p)e^t}\right)^r$ | Beta |
| Poisson $(\lambda)$ | $e^{-\lambda}\lambda^k/k!$ | $\lambda$ | $\lambda$ | $\exp(\lambda(e^t-1))$ | Gamma (rate) |
| Uniform (a,b) | $\dfrac{1}{b-a}$ | $\dfrac{a+b}{2}$ | $\dfrac{(b-a)^2}{12}$ | $\dfrac{e^{tb}-e^{ta}}{t(b-a)}$ | — |
| Exponential $(\lambda)$ | $\lambda e^{-\lambda x}$ | $1/\lambda$ | $1/\lambda^2$ | $\dfrac{\lambda}{\lambda-t}\;(t<\lambda)$ | Gamma |
| Normal $(\mu,\sigma^2)$ | $\dfrac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$ | $\mu$ | $\sigma^2$ | $\exp(\mu t+\tfrac12\sigma^2t^2)$ | Normal (mean), inverse-gamma (variance) |
| Gamma $(k,\theta)$ | $\dfrac{x^{k-1}e^{-x/\theta}}{\theta^k\Gamma(k)}$ | $k\theta$ | $k\theta^2$ | $(1-\theta t)^{-k}\;(t<1/\theta)$ | Gamma (Poisson rate), inverse-gamma (normal variance) |
| Beta (a,b) | $\dfrac{x^{a-1}(1-x)^{b-1}}{B(a,b)}$ | $\dfrac{a}{a+b}$ | $\dfrac{ab}{(a+b)^2(a+b+1)}$ | no simple closed form | Binomial / Bernoulli |
| Student-t $(\nu)$ | $\dfrac{\Gamma((\nu+1)/2)}{\sqrt{\nu\pi}\,\Gamma(\nu/2)}\left(1+\tfrac{x^2}{\nu}\right)^{-(\nu+1)/2}$ | $0\;(\nu>1)$ | $\dfrac{\nu}{\nu-2}\;(\nu>2)$ | undefined | — |
| Chi-squared (k) | $\dfrac{x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma(k/2)}$ | k | 2k | $(1-2t)^{-k/2}\;(t<1/2)$ | — |
| Cauchy $(x_0,\gamma)$ | $\dfrac{1}{\pi\gamma\left[1+\left(\frac{x-x_0}{\gamma}\right)^2\right]}$ | undefined | undefined | undefined | — |
| Lognormal $(\mu,\sigma^2)$ | $\dfrac{1}{x\sigma\sqrt{2\pi}}e^{-(\ln x-\mu)^2/2\sigma^2}$ | $e^{\mu+\sigma^2/2}$ | $(e^{\sigma^2}-1)e^{2\mu+\sigma^2}$ | undefined | — |
Quantiles and cdfs are computed by Prob.dist in assets/js/prob-viz.js; the regularised incomplete gamma and beta functions behind the gamma, chi-squared, Poisson, binomial, beta and Student-t cdfs are in Prob.sf. The conjugate-prior column is the reason a Bayesian update stays in closed form — the posterior lands back in the same family as the prior.
Where to go next
From foundations to algorithms
This volume ends at the limit theorems. The algorithms those theorems justify — maximum likelihood, intervals, sampling, Markov chains, MCMC, and the Bayes, Kalman and particle filters — are the companion volume, Probability in Action, Interactively. Its own routing table maps each of those ideas to the AI, vision and robotics page that consumes it.