List outcomes first; only then assign probabilities to events.
A condition restricts the sample space to outcomes still compatible with the evidence.
Bayes’ theorem turns an easier generative direction into the inferential direction we need.
Decide whether order matters and repetition is allowed before choosing a formula.
A random variable maps outcomes to numbers; expectation is a weighted average, not necessarily an attainable value.
Expectation compresses a distribution into a weighted average; LOTUS handles transformed values, while linearity makes sums tractable even without independence.
Variance describes a random variable's spread; Poisson models counts in a fixed interval and approximates a large-n, small-p binomial.
A continuous variable assigns zero probability to a point and area to intervals; CDFs, Uniform, Exponential, and memorylessness build on that distinction.
Standardization maps Normal variables to Z; Phi, linear transforms, and continuity correction turn intervals and large binomials into computable probabilities.
A joint distribution retains the full relationship among variables; marginals, conditionals, independence, and Bayes extract different answers from it.
Inference multiplies each hidden-variable prior by an observation likelihood and normalizes; the same loop handles repeated evidence and discretized continuous beliefs.
A Bayesian network factorizes a huge joint through conditional independence; ancestral sampling generates joint samples, and rejection sampling filters them into a conditional.
The Multinomial extends two-category Binomial counts to many categories; the same PMF models documents as word counts for Bayesian authorship with log-scores.
A Beta distribution represents full belief about an unknown success rate; success/failure data updates two parameters for posteriors, smoothing, and Thompson-sampling decisions.
A few independent sums have closed forms; general IID sums become approximately Normal under the CLT, with continuity correction for discrete sums.
The bootstrap treats a sample histogram as a population proxy, resampling with replacement to approximate a statistic's sampling distribution, error bar, or null p-value.
Expected cost in randomized code can be conditioned on the first random choice; counting problems become indicator sums, often avoiding the full distribution entirely.
Surprise turns rare events into bits; entropy is expected surprise, information gain selects uncertainty-reducing questions, and KL measures excess cost from a model distribution.
MLE fixes observed data and optimizes parameters; log-likelihood turns products into sums, but a maximum can also lie on a boundary.
Logistic regression turns a linear score into a Bernoulli probability with sigmoid; the gradient xⱼ(y-ŷ) follows directly from the log-likelihood chain rule.
Classifier comparison requires held-out data, baselines, calibration, precision/recall, and an explicit fairness criterion—not accuracy alone.
A neural network stacks logistic units; a forward pass computes probabilities, while backpropagation reuses output error to obtain every gradient.