Advanced probability, moments and conditional models
A probability model requires a sample space, events and clearly stated assumptions about selection or dependence. Independence is stronger than uncorrelatedness and cannot be inferred merely from a zero covariance. Conditional probabilities change the information set; Bayes' rule combines a prior probability with likelihood information, while the law of total probability accounts for mutually exclusive exhaustive cases. A mathematically valid answer can still be irrelevant if it conditions on the wrong event. Check denominators and whether the conditioning event has positive probability.
Expected value is linear whenever the relevant expectations exist, without requiring independence. Variance of a linear combination includes covariance terms. For example, if Var(X)=9, Var(Y)=16and Cov(X,Y)=−6, then Var(2X−Y)=4×9+16−4×(−6)=76. The correlation is−6/(3×4)=−0.5. Omitting covariance would change the result. A covariance matrix must be positive semidefinite, so not every arbitrary collection of pairwise numbers describes a valid joint model.
Distribution families encode different mechanisms. Independent Poisson counts add to a Poisson count with summed rates. A binomial model requires a fixed number of independent equal-probability Bernoulli trials under its usual definition; sampling without replacement can instead produce a hypergeometric model. An exponential waiting-time model has a particular memoryless property and rate convention. State whether a normal parameter denotes standard deviation or variance, and distinguish a density height from a probability. Integrals and sums must cover the actual support.
Inequalities provide bounds rather than exact probabilities. Markov's inequality applies to a nonnegative variable with finite mean. Chebyshev's inequality uses a finite variance and bounds deviations from the mean without assuming normality. If standard deviation is5, the probability of a deviation of at least15is at most1/9under that bound. A normal-model tail probability, if justified, would be a different calculation. A loose bound can still be mathematically correct; do not mistake its size for evidence that it is the actual event frequency.
Limit theorems have assumptions and distinct modes of convergence. Laws of large numbers concern empirical averages approaching an expectation under specified conditions. Central limit theorems describe a scaled distributional limit, not a claim that every finite sample is exactly normal or that all heavy-tailed models have finite variance. Estimator bias, variance, consistency and an interval's repeated-sampling coverage answer different questions. A frequentist confidence level is not automatically a posterior probability for a fixed unknown parameter. A hypothesis-test p-value is conditional on the null model, not the probability that the null hypothesis is true. Advanced problems reward explicit model selection and coherent reasoning, including recognition when the supplied information is insufficient.