Probability and Statistics Refresher

Use complementary counting to simplify constrained counting problems.

Construct a bijection to prove a simple combinatorial identity.

Apply the pigeonhole principle to establish existence claims.

Count nonnegative and positive integer solutions using stars and bars.

Apply inclusion–exclusion to count outcomes satisfying overlapping conditions.

Formulate and solve a simple recurrence relation for a counting sequence.

Run Python code in a notebook and use variables, numeric types, and arithmetic expressions.

Use lists, ranges, conditionals, and loops to implement finite calculations.

Write and call functions that accept inputs and return numerical results.

Use NumPy arrays for vectorized numerical calculations.

Represent sample spaces and events using set notation and set operations.

Compute event probabilities from the probability axioms, complements, and unions.

Compute and interpret conditional probabilities.

Apply the multiplication rule and the law of total probability.

Update probabilities using Bayes’ theorem.

Test events for independence and distinguish pairwise, mutual, and conditional independence.

Compute probabilities in finite equally likely sample spaces by counting favourable outcomes.

Model sampling problems correctly according to whether order matters and replacement is allowed.

Compute exact probabilities for sampling without replacement using hypergeometric counting.

Compute probabilities in elementary occupancy problems.

Compute collision probabilities, including birthday-type problems.

Solve matching and derangement probability problems using inclusion–exclusion.

Use Python's math and itertools libraries to enumerate finite outcomes and validate a hand-derived probability.

Represent a discrete random variable using its probability mass function and cumulative distribution function.

Compute expectations of functions of a discrete random variable.

Compute and interpret the variance and standard deviation of a discrete random variable.

Derive binomial probabilities from independent Bernoulli trials.

Compute waiting-time probabilities using the geometric distribution.

Model rare-event counts using the Poisson distribution.

Use indicator variables and linearity of expectation to calculate expected counts.

Compute probabilities from continuous densities and cumulative distribution functions, including uniform and piecewise-density examples.

Compute the expectation and variance of a continuous random variable.

Use the exponential distribution and verify its memoryless property.

Standardize normal random variables and calculate normal probabilities and quantiles.

Derive the distribution of a monotone transformation of a continuous random variable.

Define joint probability mass and density functions and evaluate probabilities over two-dimensional support regions.

Obtain marginal distributions from a joint distribution.

Compute a conditional probability mass function or density from a joint distribution.

Determine whether random variables are independent by factorizing their joint distribution and checking support conditions.

Compute and interpret covariance and correlation.

Compute a conditional expectation from a joint distribution.

Apply the law of total expectation.

Apply the law of total variance.

Derive the distribution of a sum using convolution.

Use the law of large numbers to explain convergence of sample averages.

Standardize sums and sample means using the central limit theorem.

Use a normal approximation with a continuity correction for a discrete probability.

Use Student's t distribution to standardize a sample mean when the population variance is unknown and compute critical values.

Generate reproducible random samples with NumPy.

Use NumPy and Matplotlib to summarize an empirical distribution with numerical summaries, a histogram, and an empirical cumulative distribution function.

Estimate a probability by Monte Carlo simulation and quantify its simulation uncertainty with a Monte Carlo standard error.

Investigate the sampling distribution of a sample mean or proportion through repeated simulation.

Distinguish a population, parameter, independent random sample, statistic, estimator, and estimate.

Construct a likelihood function from an independent random sample.

Derive maximum likelihood estimators for Bernoulli and Poisson models.

Derive maximum likelihood estimators for the mean and variance of a normal model.

Compare estimators using bias, variance, and mean squared error.

Compute a standard error from an estimator’s sampling distribution.

Construct and correctly interpret confidence intervals for a population mean using normal and Student's t procedures.

Construct and interpret a confidence interval for a population proportion after checking its approximation conditions.

Formulate null and alternative hypotheses and identify the significance level, Type I errors, and Type II errors.

Compute and interpret a p-value from a test statistic.

Select and perform a one-sample z-test or Student's t-test for a population mean after checking its assumptions.

Select and perform a one-sample z-test or exact binomial test for a population proportion.

Perform two-sample tests for differences in means or proportions.

Perform and interpret a chi-square test for independence in a contingency table.

Calculate statistical power and determine a sample size for a simple test.

Implement standard hypothesis tests in SciPy and verify their assumptions.

Derive the ordinary least-squares coefficients for a simple linear regression.

Interpret regression coefficients and state the assumptions needed for statistical inference.

Construct a confidence interval and hypothesis test for a regression slope.

Use residual plots to diagnose nonlinearity, unequal variance, and outliers.

Distinguish association from causation using confounding and selection examples.

Estimate a treatment effect in a randomized A/B experiment, quantify its uncertainty, and explain why random assignment supports a causal interpretation.

Fit a simple regression with statsmodels and interpret its statistical output.