TEME cheat sheet

A discrete random variable takes countably many values; its probability mass function sums to 1 over all possible outcomes.

Expected value E[X] = Σ x·P(X=x); variance Var(X) = E[X²] − (E[X])², and standard deviation is the square root of variance.

The binomial distribution models the number of successes in n independent trials with success probability p; mean np, variance np(1−p).

Bayes' theorem: P(A|B) = P(B|A)·P(A) / P(B), where P(B) = Σ P(B|A_i)·P(A_i) over a partition of the sample space.

Two events are independent iff P(A∩B) = P(A)·P(B); mutually exclusive events with positive probability are never independent.

Central limit theorem: the sampling distribution of the sample mean approaches normality with mean μ and standard error σ/√n as n grows.

Type I error is rejecting a true null hypothesis (probability α); Type II error is failing to reject a false null (probability β); power = 1 − β.

A 95% confidence interval for a mean is x̄ ± z*·(σ/√n) when σ is known; z* ≈ 1.96 for 95% confidence.

A p-value is the probability, assuming the null hypothesis is true, of observing a test statistic at least as extreme as the one computed.

Correlation coefficient r lies in [−1, 1]; r² gives the proportion of variance in the response explained by a linear model.

In simple linear regression, the slope estimate b = r·(s_y/s_x); the intercept is ȳ − b·x̄.

Sensitivity = TP/(TP+FN); specificity = TN/(TN+FP); a test with high sensitivity has few false negatives.

A confidence interval excludes the null value if and only if the corresponding two-sided hypothesis test rejects the null at the same α level.

The multiplication rule for dependent events: P(A∩B) = P(A)·P(B|A); for independent events P(A∩B) = P(A)·P(B).

Increasing sample size narrows confidence intervals proportionally to 1/√n and increases statistical power without changing α.