Foundations

On sigma-finite measure spaces, a nonnegative product-measurable function has equal product and iterated integrals, possibly infinite.

A topological space is connected when it cannot be partitioned into two nonempty disjoint open sets.

A metric space is complete when every Cauchy sequence has a limit in that space.

A self-map T of a nonempty complete metric space with d(Tx,Ty)≤k d(x,y), 0≤k<1, has one fixed point.

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.

For continuous real f with f(a)f(b)<0, test the midpoint and retain a half interval with opposite endpoint signs, unless the midpoint is a root.

At distinct nodes t₀,…,tₙ, p(x)=Σyₖℓₖ(x), ℓₖ(x)=∏_(i≠k)(x−tᵢ)/(tₖ−tᵢ), is the unique interpolant of degree at most n.

A differentiable function has zero gradient at an interior local extremum.

For C¹ real f,g, a local constrained extremum of f on g(x)=c at a point where ∇g≠0 must satisfy ∇f=λ∇g together with g=c.

A differentiable function on an open convex domain is convex exactly when f(y)≥f(x)+∇f(x)·(y−x) for all domain points.

For differentiable convex f₀,fᵢ on Rⁿ and affine equalities Ax=b, feasible x is optimal if λ≥0, λᵢfᵢ(x)=0 and ∇f₀(x)+Σλᵢ∇fᵢ(x)+Aᵀν=0.

For a smooth rapidly decreasing f whose derivatives also decrease faster than any power, define f̂(ξ)=∫Rⁿe⁻ⁱˣ·ξf(x)dx.

For y″+p(x)y′+q(x)y=f(x), use independent homogeneous solutions y₁,y₂ and set yₚ=u₁y₁+u₂y₂.

An n-element set has 2ⁿ subsets; its k-element subsets correspond to length-n binary strings with k ones.

For aₙ=saₙ₋₁+taₙ₋₂ with t≠0, distinct roots r₁,r₂ of r²−sr−t give aₙ=Ar₁ⁿ+Br₂ⁿ; a repeated root r gives aₙ=(A+Bn)rⁿ.

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.