Concepts

84 atomic notes, each shared by several question types.

Base Rates & the Base-Rate FallacyThe prior probability P(A) is the base rate. When it is extreme, an accurate test still produce…2 types · 3 questions

Bayes' TheoremFor a partition A₁, …, A_k and an observed event B with P(B) > 0, P(AⱼB) = P(Aⱼ)P(BAⱼ) / Σᵢ P(A…2 types · 12 questions

Bernoulli Random VariableAn rv with only two possible values, 0 and 1. Equivalently, the indicator of an event A: I_A = …2 types · 2 questions

Binomial Approximation to the HypergeometricWhen the sample is a small fraction of the population, Hypergeometric(n, M, N) ≈ Bin(n, M/N) pr…1 types · 3 questions

Binomial cdf and Table A.1B(x; n, p) = Σ_{y ≤ x} b(y; n, p) Range translations: at most k → B(k); exactly k → B(k) − B(k−…4 types · 11 questions

Binomial DistributionX ~ Bin(n, p) counts successes in n fixed, independent, identical trials. b(x; n, p) = C(n,x)·p…8 types · 25 questions

Binomial Mean and VarianceE(X) = np V(X) = np(1−p) Shortest derivation: X = I₁ + … + Iₙ, a sum of Bernoulli indicators, s…6 types · 13 questions

Binomial pmfb(x; n, p) = C(n,x)p^x(1−p)^(n−x) Read it as (number of arrangements) × (probability of one arr…4 types · 11 questions

Capacity ConstraintsWhen a finite supply serves two complementary demands, "everyone is satisfied" becomes a two-si…1 types · 4 questions

Chebyshev's InequalityFor any distribution with finite mean μ and finite SD σ, and any k ≥ 1, P(X − μ ≥ kσ) ≤ 1/k² Bo…1 types · 2 questions

CombinationsC(n, k) = n!/(k!(n−k)!) The number of unordered subsets of size k from n distinct objects. Key …6 types · 12 questions

Complement RuleP(A) = 1 − P(A′) Trivial as algebra, decisive as strategy. Use it whenever A′ is easier to desc…13 types · 46 questions

Conditional IndependenceA and B are conditionally independent given C if P(A ∩ B C) = P(AC)·P(BC). This is what "the tw…1 types · 1 questions

Conditional ProbabilityFor P(B) > 0, P(AB) = P(A ∩ B) / P(B) Conditioning shrinks the sample space to B and renormalis…6 types · 19 questions

Continuous Random VariableAn rv whose possible values fill an interval (or a union of intervals) of the real line, and fo…1 types · 1 questions

Cumulative Distribution Function (cdf)F(x) = P(X ≤ x) = Σ_{y ≤ x} p(y) For a discrete rv, F is a step function: flat between support …5 types · 9 questions

De Morgan's Laws(A ∪ B)′ = A′ ∩ B′ ("not (A or B)" = "neither A nor B") (A ∩ B)′ = A′ ∪ B′ ("not (A and B)" = "…3 types · 6 questions

Decision Rules on a SampleA rule that converts an observed count into an action ("accept the batch if X ≤ 2"; "campaign f…1 types · 1 questions

Defining 'Success' CorrectlyA binomial model needs a well-defined trial and a constant success probability p. When p is not…1 types · 5 questions

DerangementsA derangement is a permutation with no fixed points — nobody gets their own item. The counts ar…1 types · 1 questions

Deriving the Poisson from its PostulatesLet P_k(t) = P(exactly k events in (0, t)). Decomposing (0, t + Δt) into (0, t) and (t, t + Δt)…0 types · 1 questions

Detecting DependenceCompare P(A ∩ B) with P(A)P(B): - equal → independent - P(A∩B) > P(A)P(B) → positively associat…2 types · 7 questions

Discrete Random VariableAn rv whose set of possible values is finite or countably infinite (listable in a sequence with…2 types · 10 questions

Discrete Uniform Distributionp(x) = 1/n for x = 1, 2, …, n. E(X) = (n+1)/2 V(X) = (n² − 1)/12 Derived with Σx = n(n+1)/2 and…2 types · 2 questions

Equally Likely OutcomesIf 𝒮 has N outcomes all with probability 1/N, then for any event A P(A) = N(A)/N The equal-lik…4 types · 9 questions

EventA subset of the sample space 𝒮. An event occurs when the observed outcome belongs to it. A sim…1 types · 7 questions

Existence of an ExpectationFor an infinite support, E(X) = Σ x p(x) exists only if the series converges absolutely. The st…1 types · 1 questions

Expected ValueE(X) = μ = Σ_x x·p(x) A probability-weighted average of the possible values — the long-run aver…6 types · 19 questions

Expected Value as a Decision CriterionTo choose among actions with uncertain payoffs, compute E[payoff] under each and pick the large…2 types · 4 questions

Expected Value of a FunctionE[h(X)] = Σ_x h(x)·p(x) Apply h to the values, keep the original probabilities. You never need …4 types · 10 questions

Factorial MomentsThe second factorial moment is E[X(X − 1)]. Since X(X−1) = X² − X, linearity gives E(X²) = E[X(…1 types · 1 questions

Finite-Population CorrectionThe factor (N − n)/(N − 1) multiplying the binomial variance to give the hypergeometric varianc…2 types · 5 questions

Functions of a Random VariableIf X is an rv and h is a function, then h(X) is another rv on the same sample space. To find it…1 types · 3 questions

Geometric DistributionThe negative binomial with r = 1: trials continue until the first success. Trials parameterisat…4 types · 6 questions

Heavy-Tailed DistributionsDistributions whose tails decay slowly enough that some moments fail to exist. Power-law pmfs l…1 types · 1 questions

Hypergeometric DistributionDraw n items without replacement from a population of N containing exactly M successes. Then h(…4 types · 10 questions

Inclusion–Exclusion (Three Events)P(A∪B∪C) = P(A)+P(B)+P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C) Add the singles, subtract the p…2 types · 8 questions

IndependenceA and B are independent iff any (hence all) of these hold: P(A ∩ B) = P(A)·P(B) P(AB) = P(A) P(…12 types · 32 questions

Joint Probability TableA rectangular array whose cells give P(row category ∩ column category) for two cross-classified…1 types · 5 questions

Law of the Unconscious StatisticianThe formal name for E[h(X)] = Σ h(x)p(x): you can compute the expectation of h(X) without ever …1 types · 1 questions

Law of Total ProbabilityFor a partition A₁, …, A_k of 𝒮 and any event B, P(B) = Σᵢ P(Aᵢ)·P(BAᵢ) Split B according to w…4 types · 14 questions

Linear Transformation RulesE(aX + b) = a·E(X) + b V(aX + b) = a²·V(X) σ_{aX+b} = a·σ_X A shift b moves the centre but not …2 types · 9 questions

Marginal ProbabilityThe probability of one variable's category, obtained by summing the joint probabilities over al…1 types · 5 questions

Monotonicity of ExpectationIf a ≤ X ≤ b for every possible value, then a ≤ E(X) ≤ b. Proof: multiply a ≤ x ≤ b through by …1 types · 1 questions

Monotonicity of ProbabilityIf A ⊆ B then P(A) ≤ P(B). Proof: B = A ∪ (B ∩ A′) with the pieces disjoint, so P(B) = P(A) + P…3 types · 4 questions

Multiplication Rule (Chain Rule)P(A ∩ B) = P(A)·P(BA) P(A ∩ B ∩ C) = P(A)·P(BA)·P(CA ∩ B) Just the definition of conditional pr…6 types · 13 questions

Mutual IndependenceEvents A₁, …, Aₙ are mutually independent if every subset factorises: P(A_i ∩ A_j) = P(A_i)P(A_…2 types · 2 questions

Mutually Exclusive EventsEvents are mutually exclusive (pairwise disjoint) if no two of them share an outcome: A ∩ B = A…2 types · 3 questions

Negative Binomial DistributionIndependent identical trials continue until the rth success. Devore counts failures: X = number…3 types · 6 questions

Normalization ConditionΣ p(x) = 1 over all possible values. Three distinct uses: 1. Solve for an unknown constant in a…3 types · 6 questions

Operating Characteristic (OC) CurveFor an acceptance-sampling plan (sample n, accept if at most c defectives), the OC curve plots …1 types · 1 questions

Partition of the Sample SpaceEvents A₁, …, A_k partition 𝒮 if they are mutually exclusive and their union is 𝒮 — equivalen…1 types · 3 questions

PermutationsThe number of ordered arrangements of n distinct objects is n!; the number of ordered selection…2 types · 2 questions

Permutations of a MultisetWhen objects repeat, arrangements are counted by dividing out the repetitions. For a string of …1 types · 1 questions

Poisson Approximation to the BinomialAs n → ∞ and p → 0 with np → μ, b(x; n, p) → p(x; μ) with μ = np Rule of thumb: n ≥ 50 and np ≤…1 types · 4 questions

Poisson cdf and Table A.2F(x; μ) = Σ_{y ≤ x} p(y; μ) Same range translations as the binomial: at most k → F(k); exactly …3 types · 9 questions

Poisson Distributionp(x; μ) = e^(−μ)·μ^x / x!, x = 0, 1, 2, … E(X) = μ V(X) = μ σ = √μ The mean and variance being …2 types · 7 questions

Poisson Mean and VarianceE(X) = V(X) = μ, so σ = √μ Mean: drop the x = 0 term, cancel x against x!, factor out μ, re-ind…3 types · 9 questions

Poisson pmfp(x; μ) = e^(−μ)μ^x/x! Useful values: P(X = 0) = e^(−μ) and P(X ≥ 1) = 1 − e^(−μ). The latter, …4 types · 6 questions

Poisson ProcessEvents occur over time, area or volume such that: 1. the count in a region of size t is Poisson…2 types · 7 questions

Poisson ThinningIf events arrive as a Poisson process with rate α and each is independently retained with proba…1 types · 1 questions

Probability HistogramA bar (or line) graph with a rectangle of height p(x) centred at each possible value x. The tot…1 types · 2 questions

Probability Mass Function (pmf)For a discrete rv X, p(x) = P(X = x). A function is a legitimate pmf iff (i) p(x) ≥ 0 for all x…7 types · 23 questions

Product Rule for CountingIf a task is a sequence of k stages with n₁, n₂, …, n_k choices respectively (each independent …2 types · 2 questions

Random VariableA function X : 𝒮 → ℝ assigning a number to each outcome. Random variables are typically many-t…2 types · 10 questions

Rare EventsThe regime of large n and small p, where np stays moderate. Individually improbable events beco…1 types · 3 questions

Sample SpaceThe set 𝒮 of all possible outcomes of an experiment. Choosing 𝒮 is a modelling decision, not …4 types · 13 questions

Sampling Without ReplacementItems are drawn one at a time and not returned, so successive draws are dependent: the composit…3 types · 8 questions

Sensitivity & SpecificityFor a diagnostic test on a condition D: - sensitivity = P(+ D) — the true-positive rate - speci…1 types · 2 questions

Series & Parallel SystemsWith independent components of reliability pᵢ: series (needs all): P(works) = Π pᵢ parallel (ne…1 types · 4 questions

Set Operations on EventsThree operations generate everything: - A ∪ B — at least one of A, B occurs - A ∩ B — both occu…2 types · 7 questions

Shortcut Formula for VarianceV(X) = E(X²) − [E(X)]² Proof: E[(X−μ)²] = E(X² − 2μX + μ²) = E(X²) − 2μ·μ + μ² = E(X²) − μ². Co…3 types · 7 questions

Standard Deviationσ = √V(X) Same units as X, which makes it interpretable. Used to express distances from the mea…4 types · 5 questions

Stopping-Rule ExperimentsExperiments that continue until a condition is met, so the number of trials is random and outco…4 types · 8 questions

The 'At Least One' RuleFor n independent trials each succeeding with probability p: P(at least one success) = 1 − (1 −…2 types · 5 questions

Tree DiagramA branching representation of a multi-stage experiment. Each node's outgoing branches carry the…3 types · 13 questions

Type I and Type II Errors- Type I: rejecting a true claim. P(reject H₀ true). - Type II: failing to reject a false claim…1 types · 1 questions

VarianceV(X) = σ² = E[(X − μ)²] = Σ (x − μ)²·p(x) The probability-weighted average squared deviation fr…6 types · 12 questions

Venn DiagramsA rectangle for 𝒮 with overlapping circles for events. Two events partition 𝒮 into 4 regions;…3 types · 8 questions

Waiting Times in a Poisson ProcessIn a Poisson process with rate α, the time between consecutive events has mean 1/α. Conversely,…1 types · 1 questions