Probability

Specify outcomes fine enough to describe a finite experiment.

Represent an event as a subset of a sample space.

Check a finite probability assignment for nonnegativity and unit total.

Compute an event probability by summing disjoint outcome masses.

Compute a two-event union probability with overlap correction.

Determine whether two events are mutually exclusive.

Compute an event probability within a positive-probability conditioning event.

Compute an intersection using a conditional probability.

Verify that a collection of events partitions the sample space.

Compute an event probability across an exhaustive disjoint case split.

Reverse a conditional probability using a prior and total evidence probability.

Explain a posterior result using the prevalence of the hypothesized event.

Check independence using the product condition.

Check independence within a stated conditioning event.

Define a numerical random variable on a finite sample space.

Construct a discrete random variable's probability mass function.

Compute a discrete cumulative distribution at a threshold.

Compute P(a < X ≤ b) by subtracting supplied CDF values at the endpoints.

Compute a finite random variable's expectation.

Compute the expectation of a function of a discrete random variable.

Compute the expectation of a linear combination without assuming independence.

Use an event indicator's expectation to compute a count's expected value.

Compute variance as expected squared deviation from the mean.

Express spread in the original variable's units.

Compute variance after a scalar shift and scale.

Model a binary outcome with a Bernoulli distribution.

Interpret a supplied area under a continuous density curve as an interval probability.

Identify a normal model's location and scale parameters.

Convert a normal threshold to standard-normal units.

Compute a normal interval probability from a supplied standard-normal CDF.

Construct a joint discrete distribution over paired values.

Marginalize a joint mass table over one variable.

Compute a conditional distribution from a joint mass table.

Check factorization of a discrete joint distribution into its marginals.

Identify a chi-squared variable as a sum of squared independent standard-normal variables.

Interpret a supplied standard-normal to scaled-chi-squared ratio as a t variable under independence.

Compute an expectation of a function using a finite joint table.

Compute covariance from a joint distribution.

Compute variance of a sum with a supplied covariance.

Compute a conditional mean from a conditional mass function.

Recover an unconditional mean by averaging conditional means.

Decompose variance into within-case and between-case components.

Assemble the covariance matrix of a finite random vector.

Check the identical-distribution and independence assumptions in a sampling story.

Compute the variance of an IID sample mean with finite variance.

Bound a nonnegative variable's upper tail from its mean.

Bound a deviation probability using finite variance.

Interpret convergence of an IID sample average under a stated finite-variance assumption.

Form a normal approximation for a standardized IID sum with finite positive variance.

Diagnose a normal approximation undermined by the sampling assumptions or heavy tails.

Estimate an expectation with a supplied set of independent simulated draws.

Compute the information associated with a positive-probability event.

Compute entropy of a finite probability distribution.

Compute expected negative log probability under a supplied target distribution.

Compute a finite-distribution KL divergence with valid support handling.