Probability
Checkpoints
- Convolution of Distributions Essential The distribution of the sum X + Y of two independent random variables is the convolution of their distributions. This checkpoint derives the convolution formula — (f_X * f_Y)(z) = ∫ f_X(x) f_Y(z − x) dx for densities, and the analogous sum for PMFs — applies it to compute Binomial(m, p) + Binomial(n, p) = Binomial(m + n, p), Poisson(λ) + Poisson(μ) = Poisson(λ + μ), and N(μ_1, σ_1²) + N(μ_2, σ_2²) = N(μ_1 + μ_2, σ_1² + σ_2²), and connects the convolution operation to the multiplicative behaviour of moment generating functions.
- Expectation Essential The expectation of a random variable X is its Lebesgue integral against the underlying probability measure, E[X] = ∫ X dP, equivalently ∫ x dF(x) in terms of the distribution. This checkpoint defines E[X] for discrete and absolutely continuous variables, establishes the linearity E[aX + bY] = aE[X] + bE[Y], and proves the law of the unconscious statistician E[g(X)] = ∫ g(x) dF(x).
- Moment Generating Function Essential The moment generating function M_X(t) = E[e^(tX)] encodes the moments of X as the coefficients of its Taylor expansion at t = 0: M_X^(k)(0) = E[X^k]. This checkpoint defines the MGF, computes it for the standard distributions, derives the multiplicative property M_{X+Y} = M_X · M_Y for independent X, Y, and discusses the existence and uniqueness conditions under which the MGF determines the distribution.
- Introduction to Probability Essential Modern probability is the study of measure spaces whose total mass is one — every probabilistic statement reduces to a statement about a measurable set, a measurable function, or an integral. This checkpoint motivates the measure-theoretic framework, surveys the road ahead (sample spaces, axioms, random variables, expectation, conditional structures, Markov chains), and explains why the Lebesgue integral is the right foundation.
- Moments Essential The k-th moment of a random variable X is E[X^k], and the k-th central moment is E[(X − E[X])^k]. This checkpoint defines raw and central moments, recovers mean and variance as the first moment and second central moment, surveys skewness and kurtosis as standardised third and fourth central moments, and explains in what sense moments characterise a distribution (and when they fail to).
- The Probability Axioms Essential A probability space is a measure space (Ω, ℱ, P) in which the measure P assigns mass 1 to the whole sample space. This checkpoint states Kolmogorov's three axioms — non-negativity, normalisation, and countable additivity — derives the immediate consequences (monotonicity, inclusion–exclusion, continuity from below and above), and shows that probability is just measure theory with a unit total mass constraint.
- Sample Space and Events Essential A sample space Ω is the set of all possible outcomes of a random experiment, and an event is a subset of Ω. This checkpoint fixes the vocabulary — outcome, sample space, event, elementary event, complement, union, intersection — and shows how set-theoretic operations on events mirror logical operations on propositions.
- Variance Essential The variance Var(X) = E[(X − E[X])²] measures the average squared deviation of X from its mean, and is the simplest non-trivial statistic of spread. This checkpoint derives the computational form Var(X) = E[X²] − (E[X])², proves the scaling rule Var(aX + b) = a² Var(X), and discusses why variance — not standard deviation — is the algebraically natural quantity even though only the standard deviation σ = √Var(X) has the same units as X.