Foundations

A probability assigns nonnegative, normalised, countably additive weights to events.

Conditioning restricts attention to an event of positive probability.

Expected value weights possible values by their probabilities.

A confidence level describes repeated-sampling coverage of an interval procedure.

A distribution assigns probabilities to a random variable's values.

A normal model is specified by location and positive scale.

Different count mechanisms require different distributions.

A sample mean concentrates near its mean under stated assumptions.

Centered sample averages have an asymptotic normal law.

A conditional distribution renormalizes a joint law.

Likelihood is a function of a parameter given observed data.

A p-value is calibrated under a specified null model.

Selection and treatment assignment answer different questions.

Distribution-free procedures still require assumptions.

For square-integrable real X,Y, Var X=E[(X−EX)²] and Cov(X,Y)=E[(X−EX)(Y−EY)].

A joint mass or density specifies simultaneous values; marginals sum or integrate out the other coordinates.

E[X|Y=y] averages X using its conditional law.

For square-integrable X, Var(X)=E[Var(X|Y)]+Var(E[X|Y]).

M_X(t)=E[eᵗˣ] is an MGF when it is finite on an open interval about zero.

In the shape–rate convention Gamma(α,β) has density βᵅxᵅ⁻¹e⁻ᵝˣ/Γ(α) for x>0, α,β>0.

For estimator T of θ, bias is EθT−θ and MSE is Eθ[(T−θ)²]=Varθ(T)+bias².

An estimator sequence Tₙ is consistent for θ when Pθ(|Tₙ−θ|>ε)→0 for every ε>0.

For a differentiable scalar iid model with score sθ=∂θlog fθ(X), Fisher information is I(θ)=Eθ[sθ²].

For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹.

With independent Bernoulli Yᵢ, logistic regression assumes log[pᵢ/(1−pᵢ)]=xᵢᵀβ.

Residuals eᵢ=yᵢ−f(xᵢ;β̂) compare observed and fitted responses.

A chain satisfies P(Xₙ₊₁=j|X₀,…,Xₙ)=Pᵢⱼ when Xₙ=i, where P has nonnegative entries and each row sums to one.

A stationary distribution is a nonnegative row vector π with Σπᵢ=1 and πP=π.

A rate-λ Poisson counting process starts at zero, has independent increments and N(t+s)−N(s) distributed Poisson(λt).

Standard Brownian motion has B(0)=0, almost surely continuous paths and independent increments with B(t)−B(s)∼N(0,t−s) for 0≤s<t.