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1890 learning items

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Quiz

Limits, Continuity & Derivatives

Build the single-variable calculus needed for later probability densities and estimation. Interpret limits, continuity and instantaneous rates, and apply power, product, quotient, chain, exponential and logarithmic differentiation rules. Prerequisites: algebra, functions and the preceding discrete probability topics. Diagnostic questions distinguish a limit from a point value and local behavior from global claims. Further reading: Paul Dawkins: Continuity Paul Dawkins: Differentiation formulas Paul Dawkins: Chain rule Cover: Photo by Pok Rie on Pexels.

statistics

Oct 23, 2026

20 questions

Quiz

Multinomial Distribution

Extend binary trials to several mutually exclusive categories. Study the joint count formula, binomial marginals, merged categories, negative covariance, conditional renormalization and fixed-total constraints. Prerequisites: binomial and Poisson models, elementary counting, expectations and covariance. Worked examples show why independent trials do not imply independent category totals. Further reading: Kyle Siegrist: Multinomial distribution Cover: Photo by Ilo Frey on Pexels.

statistics

Oct 23, 2026

15 questions

Quiz

Poisson Distribution & The Poisson Process

Connect Poisson counts to event arrivals over time. Practice rate conversions, exact probabilities, moments, independent streams and rare-event approximations, then diagnose overlapping windows, batch arrivals and changing rates. Prerequisites: discrete random variables, independence, binomial counts and conditional probability. More demanding items derive conditional splits and mixture variability from elementary probability identities. Further reading: Hossein Pishro-Nik: Poisson process Hossein Pishro-Nik: Special discrete distributions Cover: Photo by Kostiantyn Zavhorodnii on Pexels.

statistics

Oct 23, 2026

35 questions

Quiz

Hypergeometric Distribution

Model fixed-size random samples from finite populations without replacement. Explore feasible counts, combinatorial probabilities, expectations, negative dependence, the finite-population correction and binomial limits. Prerequisites: combinations, binomial models and variance. Applied cases examine quality inspections, complementary samples, selection bias and why matching a mean does not establish a model. Further reading: Kyle Siegrist: Hypergeometric distribution Cover: Photo by Bruno Scramgnon on Pexels.

statistics

Oct 23, 2026

25 questions

Quiz

Geometric & Negative Binomial Distributions

Understand geometric and negative-binomial waiting times with explicit trials-versus-failures conventions. Practice exact probabilities, tail events, memorylessness, expected waits, variability, conditional progress and retry limits. Prerequisites: independent Bernoulli trials, binomial counts and basic moments. Challenges examine changing success chances and hidden mixtures without relying on untaught calculus. Further reading: Kyle Siegrist: Geometric distribution Kyle Siegrist: Negative binomial distribution Cover: Photo by Jordan Benton on Pexels.

statistics

Oct 23, 2026

30 questions

Quiz

Binomial Distribution

Choose and analyze binomial models for fixed collections of binary trials. Work from assumptions and simple counts to exact probabilities, cumulative tables, indicator derivations, sample proportions, parameter recovery and design diagnostics. Prerequisites: probability rules, independence, expectation, variance and Bernoulli indicators. Worked solutions distinguish valid models from heterogeneous, dependent or stopped experiments and explain the reasoning behind each distractor. Further reading: Kyle Siegrist: Binomial distribution Hossein Pishro-Nik: Special discrete distributions Cover: Photo by CocaKolaLips on Pexels.

statistics

Oct 23, 2026

35 questions

Quiz

Discrete Uniform & Bernoulli Distributions

Compare equal-mass finite models with zero-one event indicators. Compute means and variances, inspect Bernoulli CDFs, handle endpoint parameters and analyze transformed rewards or spaced supports. Prerequisites: PMFs, expectation and variance. Challenges remain elementary and focus on model assumptions, parameter ambiguity and consequences of coding. Includes single-select and multiselect questions with worked solutions and distractor feedback. References: [Notes](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.bernoulli.html), [Notes](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.randint.html), [3.2.4 Definition, computational formula and affine transformation](https://www.probabilitycourse.com/chapter3/3_2_4_variance.php), [3.2.2 Definition and linearity theorem](https://www.probabilitycourse.com/chapter3/3_2_2_expectation.php).

statistics

Oct 23, 2026

15 questions

Quiz

Variance, Standard Deviation & Moments of Discrete RVs

Calculate discrete variance, standard deviation and raw or central moments from finite probability models. Explore transformations, degenerate cases, moment consistency, squared error and distributions with equal variance but different higher moments. Prerequisites: expected value and LOTUS. Worked explanations distinguish known-distribution calculations from sample estimation. Includes single-select and multiselect questions with worked solutions and distractor feedback. References: [3.2.4 Definition, computational formula and affine transformation](https://www.probabilitycourse.com/chapter3/3_2_4_variance.php), [3.2.3 Functions and LOTUS](https://www.probabilitycourse.com/chapter3/3_2_3_functions_random_var.php).

statistics

Oct 23, 2026

25 questions

Quiz

Expected Value, LOTUS & Linear Properties

Compute expected costs, rewards and transformed quantities using weighted sums and LOTUS. Apply linearity with dependent variables, diagnose invalid nonlinear substitutions, and compare models with equal marginal means. Prerequisites: discrete PMFs, basic algebra and powers. A supplied-series fact supports the final expectation-existence challenge without requiring calculus. Includes single-select and multiselect questions with worked solutions and distractor feedback. References: [3.2.2 Definition and linearity theorem](https://www.probabilitycourse.com/chapter3/3_2_2_expectation.php), [3.2.3 Functions and LOTUS](https://www.probabilitycourse.com/chapter3/3_2_3_functions_random_var.php).

statistics

Oct 23, 2026

25 questions

Quiz

Discrete Random Variables, PMF & CDF

Build discrete random-variable fluency by mapping outcomes to numbers, validating PMFs and reading step-function CDFs. Practice jumps, tails, endpoint-sensitive intervals, many-to-one transformations and quantiles. Prerequisites: events and probability rules. The final questions examine what a distribution determines and what incomplete CDF information leaves unknown. Includes single-select and multiselect questions with worked solutions and distractor feedback. References: [3.1.3 Definition and properties](https://www.probabilitycourse.com/chapter3/3_1_3_pmf.php), [3.2.1 CDF definition and properties](https://www.probabilitycourse.com/chapter3/3_2_1_cdf.php), [3.2.3 Functions and LOTUS](https://www.probabilitycourse.com/chapter3/3_2_3_functions_random_var.php).

statistics

Oct 23, 2026

25 questions

Quiz

Probability Paradoxes & Classical Edge Cases

Explore why surprising probabilities depend on the experiment. Compare Monty Hall host policies, birthday collisions and reversed evidence claims, and solve small absorbing games by first-step equations. Prerequisites: counting, conditioning and Bayes rules. Finite examples are derived directly; full gambler’s ruin and stochastic-process theory are deferred. Includes single-select and multiselect questions with worked solutions and distractor feedback. References: [Monty Hall and Birthday problem](https://web.stanford.edu/class/stats60/lectures/11-lecture-coincidences.html), [Mistakes in conditional probability; prosecutor fallacy](https://web.stanford.edu/class/stats60/lectures/14-lecture-fallacies.html), [1.4.0 Definition, conditional axioms and chain rule](https://www.probabilitycourse.com/chapter1/1_4_0_conditional_probability.php), [1.4.2 Law of total probability](https://www.probabilitycourse.com/chapter1/1_4_2_total_probability.php).

statistics

Oct 23, 2026

20 questions

Quiz

Law of Total Probability & Bayes' Theorem

Practice total probability and Bayes updating using source attribution, alerts and explicitly hypothetical screening scenarios. Interpret sensitivity, specificity, predictive values and likelihood ratios, then investigate base rates, repeated evidence, missing parameters and population shifts. Prerequisites: conditional probability, independence and elementary algebra. Diagnostic examples are stipulated mathematical models. Includes single-select and multiselect questions with worked solutions and distractor feedback. References: [1.4.2 Law of total probability](https://www.probabilitycourse.com/chapter1/1_4_2_total_probability.php), [1.4.3 Bayes rule](https://www.probabilitycourse.com/chapter1/1_4_3_bayes_rule.php).

statistics

Oct 23, 2026

35 questions