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OpenStem
@openstem · Joined Jul 2026
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Flashcards10 cards
State the axioms of probability (Kolmogorov).1 / 10
1. P(A) ≥ 0 for every event A. 2. P(Ω) = 1. 3. For mutually exclusive events A₁, A₂, …: P(A₁ ∪ A₂ ∪ …) = P(A₁) + P(A₂) + …
Mathematics · L4 · Probability — Fundamentals
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Mathematics · L4 · Probability — FundamentalsFlashcards10 cards
State the Fundamental Theorem of Arithmetic.1 / 10
Every integer n > 1 is either prime or can be written as a unique product of prime numbers (up to order). Example: 360 = 2³ · 3² · 5.
Mathematics · L4 · Number Theory & Combinatorics
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Mathematics · L4 · Number Theory & CombinatoricsFlashcards11 cards
What are the three measures of central tendency?1 / 11
Mean (arithmetic average: Σx/n), Median (middle value when sorted; robust to outliers), Mode (most frequent value). For skewed distributions, the median is often preferred over the mean.
Mathematics · L4 · Descriptive Statistics — Summarising Data
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Mathematics · L4 · Descriptive Statistics — Summarising DataFlashcards10 cards
State the three Kolmogorov axioms of probability.1 / 10
1) P(A) ≥ 0 for all events A. 2) P(Ω) = 1 (the sample space has probability 1). 3) For mutually exclusive events: P(A₁ ∪ A₂ ∪ …) = ΣP(Aᵢ). All other probability rules follow from these.
Mathematics · L4 · Probability Foundations
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Mathematics · L4 · Probability FoundationsFlashcards11 cards
What is a point estimate vs an interval estimate?1 / 11
A point estimate is a single statistic (e.g., x̄) used to estimate a population parameter (μ). An interval estimate (confidence interval) provides a range [L, U] that captures the true parameter with a stated confidence level (e.g., 95%).
Mathematics · L4 · Statistical Inference — Estimation & Tests
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Mathematics · L4 · Statistical Inference — Estimation & TestsFlashcards12 cards
Write the simple linear regression model.1 / 12
Y = β₀ + β₁X + ε, where β₀ is the intercept, β₁ is the slope, and ε ~ N(0, σ²) is the error. OLS estimates β̂₀ and β̂₁ minimise the sum of squared residuals Σ(yᵢ − ŷᵢ)².
Mathematics · L4 · Regression & Predictive Modelling
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Mathematics · L4 · Regression & Predictive ModellingFlashcards10 cards
State Bayes' theorem.1 / 10
P(H|E) = P(E|H) × P(H) / P(E). Posterior ∝ Likelihood × Prior. P(H): prior belief; P(E|H): likelihood of evidence given hypothesis; P(H|E): posterior belief after observing evidence.
Mathematics · L4 · Bayesian Statistics — Thinking in Probability
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Mathematics · L4 · Bayesian Statistics — Thinking in ProbabilityFlashcards9 cards
What is an ordinary differential equation (ODE), and what does its order mean?1 / 9
An ODE relates an unknown function y(x) to its derivatives, e.g. dy/dx = 3x − y. The order is the highest derivative that appears — dy/dx = f(x,y) is first order, d²y/dx² = f(x,y,y′) is second order.
Mathematics · L4 · Differential Equations — First-Order ODEs
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Mathematics · L4 · Differential Equations — First-Order ODEsFlashcards9 cards
What is a partial derivative ∂f/∂x of f(x,y)?1 / 9
The rate of change of f with respect to x, holding y fixed — differentiate normally, treating every other variable as a constant.
Mathematics · L4 · Multivariable Calculus — Partial Derivatives & Optimization
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Mathematics · L4 · Multivariable Calculus — Partial Derivatives & OptimizationFlashcards5 cards
State the ε–N definition of convergence of a sequence (aₙ) → L.1 / 5
For every ε > 0 there exists N ∈ ℕ such that for all n ≥ N, |aₙ − L| < ε.
Mathematics · L5 · Real Analysis Foundations
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Mathematics · L5 · Real Analysis FoundationsFlashcards5 cards
Define a normal subgroup N ⊴ G.1 / 5
N is a subgroup of G such that gNg⁻¹ = N for every g ∈ G (equivalently, every left coset equals the corresponding right coset). Normal subgroups are precisely the kernels of group homomorphisms, and they allow the construction of the quotient group G/N.
Mathematics · L5 · Group & Ring Theory
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Mathematics · L5 · Group & Ring TheoryFlashcards5 cards
Define a probability space (Ω, ℱ, P).1 / 5
A probability space consists of: Ω (sample space, the set of all outcomes); ℱ (a σ-algebra on Ω, the collection of measurable events); and P (a probability measure on (Ω, ℱ) with P(Ω) = 1 and countable additivity: P(⋃ᵢ Aᵢ) = Σᵢ P(Aᵢ) for pairwise-disjoint events Aᵢ ∈ ℱ).
Mathematics · L5 · Measure-Theoretic Probability
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Mathematics · L5 · Measure-Theoretic ProbabilityFlashcards5 cards
State the Weak Law of Large Numbers (WLLN) and the Strong Law (SLLN). How do they differ?1 / 5
Let X₁, X₂, … be i.i.d. with E[X₁] = μ and (for WLLN) Var(X₁) < ∞. WLLN: X̄ₙ →ᴾ μ (convergence in probability). SLLN (Kolmogorov): with E[|X₁|] < ∞ sufficient, X̄ₙ →ᵃ·ˢ· μ (almost-sure convergence). Almost-sure is strictly stronger: ω-by-ω convergence for a set of measure 1.
Mathematics · L5 · Asymptotics & Limit Theorems
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Mathematics · L5 · Asymptotics & Limit TheoremsFlashcards9 cards
Define a topological space (X, τ).1 / 9
A set X together with a collection τ of subsets (the open sets) such that: (i) ∅, X ∈ τ; (ii) τ is closed under arbitrary unions; (iii) τ is closed under finite intersections. τ is called a topology on X.
Mathematics · L5 · Point-Set Topology
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Mathematics · L5 · Point-Set TopologyFlashcards9 cards
Define a holomorphic (complex-differentiable) function.1 / 9
f: U → ℂ (U open in ℂ) is holomorphic at z₀ if lim_{h→0} [f(z₀+h) − f(z₀)]/h exists, where h → 0 through complex values from every direction — a far stronger requirement than real differentiability.
Mathematics · L5 · Complex Analysis
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Mathematics · L5 · Complex AnalysisFlashcards8 cards
Define a convex set.1 / 8
C is convex if for all x, y ∈ C and t ∈ [0,1], tx + (1−t)y ∈ C — the entire line segment between any two points of C stays inside C.
Mathematics · L5 · Convex Optimization
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Mathematics · L5 · Convex OptimizationQuiz3 questions
What is 4 + 3?
A6B7C8D5
Mathematics · L1 · Numbers and Counting
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Mathematics · L1 · Numbers and CountingQuiz3 questions
How many sides does a triangle have?
A2B4C3D5
Mathematics · L1 · Shapes Around Us
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Mathematics · L1 · Shapes Around UsQuiz3 questions
One tally bundle (4 lines crossed by 1) equals how many?
A4B5C6D10
Mathematics · L1 · Tally Marks and Counting
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Mathematics · L1 · Tally Marks and CountingQuiz2 questions
Each picture stands for 2 votes. There are 3 pictures for apples. How many votes did apples get?
A2B3C6D5
Mathematics · L1 · Reading Pictographs
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Mathematics · L1 · Reading PictographsQuiz5 questions
Sam has 3 marbles. He finds 5 more marbles. How many marbles does he have now?
A8B7C6D9
Mathematics · L1 · Addition & Subtraction Stories
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Mathematics · L1 · Addition & Subtraction StoriesQuiz5 questions
Skip count by 2s: 2, 4, 6, __. What number is missing?
A7B8C9D10
Mathematics · L1 · Skip Counting & Patterns
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Mathematics · L1 · Skip Counting & PatternsQuiz5 questions
You compare two ribbons by lining up one end of each. Ribbon A sticks out further than Ribbon B. What can you say?
ARibbon A is shorter than Ribbon BBRibbon A is longer than Ribbon BCBoth ribbons are the same lengthDYou cannot tell without a ruler
Mathematics · L1 · Measuring Length & Weight
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Mathematics · L1 · Measuring Length & WeightQuiz5 questions
The long hand points to 12 and the short hand points to 5. What time is it?
A5 o'clockB12 o'clockCHalf past 5D6 o'clock
Mathematics · L1 · Telling Time & Calendars
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Mathematics · L1 · Telling Time & CalendarsQuiz5 questions
Which 3D shape is a dice most like?
ACubeBSphereCConeDCylinder
Mathematics · L1 · 3D Shapes & Symmetry
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Mathematics · L1 · 3D Shapes & SymmetryQuiz5 questions
You sort a box of buttons by one rule: 'is it round?'. What goes in the 'round' group?
AOnly round buttonsBAll the buttons, round or notCOnly square buttonsDNothing — buttons cannot be sorted
Mathematics · L1 · Sorting with Venn Diagrams
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Mathematics · L1 · Sorting with Venn DiagramsQuiz5 questions
Which word describes an event that will definitely happen, with no doubt at all?
AImpossibleBUnlikelyCLikelyDCertain
Mathematics · L1 · Likely, Unlikely, Certain
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Mathematics · L1 · Likely, Unlikely, CertainQuiz5 questions
You have 1 quarter, 2 dimes, and 1 nickel. How much money do you have in total?
A45¢B50¢C55¢D60¢
Mathematics · L1 · Money & Making Change
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Mathematics · L1 · Money & Making ChangeQuiz5 questions
What does 3 × 4 mean, using repeated addition?
A4 + 4 + 4B3 + 3 + 3C4 + 3D4 − 3
Mathematics · L1 · Simple Multiplication
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Mathematics · L1 · Simple MultiplicationQuiz5 questions
12 cookies are shared equally among 4 friends. How many cookies does each friend get?
A2B3C4D6
Mathematics · L1 · Simple Division
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Mathematics · L1 · Simple Division