IIT JAM MS flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Cauchy sequence

A sequence (x_n) in a metric space such that for every ε > 0 there exists N with |x_m − x_n| < ε for all m, n > N. In R, Cauchy iff convergent.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c in (a,b) with f'(c) = 0. It is the base case for the mean value theorem.

Lagrange mean value theorem

If f is continuous on [a,b] and differentiable on (a,b), then f'(c) = (f(b) − f(a))/(b − a) for some c in (a,b). It links average rate of change to an instantaneous derivative.

Rank–nullity theorem

For a linear map T: V → W with V finite-dimensional, rank(T) + nullity(T) = dim(V). Rank is dim(image) and nullity is dim(kernel).

Lagrange's theorem (groups)

For a finite group G and subgroup H, the order of H divides the order of G. Consequently the order of any element also divides |G|, so groups of prime order are cyclic.

Taylor's theorem with remainder

f(x) = Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + R_n(x), where R_n can be written as f^(n+1)(c)(x−a)^{n+1}/(n+1)! for some c between a and x.

Ratio test for series

For Σa_n with a_n ≠ 0, let L = lim |a_{n+1}/a_n|. The series converges absolutely if L < 1, diverges if L > 1, and the test is inconclusive when L = 1.

Cayley–Hamilton theorem

Every square matrix satisfies its own characteristic polynomial: p_A(A) = 0. This gives a way to express high powers of A via lower powers and to compute A^{-1}.

Eigenvalue equation

λ is an eigenvalue of A if det(A − λI) = 0, and a nonzero vector v with Av = λv is a corresponding eigenvector. The set of eigenvalues is the spectrum of A.

Diagonalizability criterion

An n×n matrix is diagonalizable over a field iff it has n linearly independent eigenvectors, equivalently the sum of geometric multiplicities of all eigenvalues equals n.

Continuity versus differentiability

Differentiability at a point implies continuity there, but continuity does not imply differentiability. For example, f(x) = |x| is continuous but not differentiable at x = 0.

L'Hôpital's rule

For a 0/0 or ∞/∞ indeterminate form, lim f/g = lim f'/g' provided the latter limit exists. The form must be verified before applying the rule.

Wallis integral recurrence

For I_n = ∫₀^{π/2} sin^n x dx, I_n = ((n−1)/n) I_{n−2} with I_0 = π/2 and I_1 = 1. The same recurrence holds for the cosine version.

Conjugate prior

A prior distribution is conjugate for a likelihood if the resulting posterior belongs to the same family. For example, a Beta prior with Binomial data yields a Beta posterior.

Unbiased estimator

An estimator θ̂ is unbiased for θ if E(θ̂) = θ for all θ. The sample variance with denominator n−1 is unbiased for σ², while the MLE with denominator n is biased.

Cramér–Rao lower bound

For an unbiased estimator θ̂ based on n iid observations, Var(θ̂) ≥ 1/(n I(θ)), where I(θ) is the Fisher information per observation. Equality holds for efficient estimators.

Sufficient statistic

A statistic T(X) is sufficient for θ if the conditional distribution of X given T does not depend on θ. By the factorization theorem, the joint density then splits as g(T;θ)h(x).

Neyman–Pearson lemma

For simple H₀ vs simple H₁, the most powerful test of size α rejects H₀ when the likelihood ratio f₁(x)/f₀(x) exceeds a threshold chosen to give size α.

Type I and Type II errors

A Type I error rejects a true H₀ (probability α); a Type II error fails to reject a false H₀ (probability β). Power equals 1 − β, the probability of correctly rejecting H₀.

p-value interpretation

The p-value is the probability, assuming H₀ is true, of observing a test statistic at least as extreme as the one computed. Reject H₀ at level α when p ≤ α.

t confidence interval for mean

With unknown normal variance, the interval for μ is X̄ ± t_{α/2,n−1} · S/√n, where S is the sample standard deviation and t uses n−1 degrees of freedom.

Chi-square distribution of S²

For iid normal data, (n−1)S²/σ² follows a chi-square distribution with n−1 degrees of freedom, and it is independent of the sample mean X̄.

Sampling distribution of X̄

If X_i are iid N(μ, σ²), then X̄ is N(μ, σ²/n). By the central limit theorem, X̄ is approximately normal for large n even when the population is not normal.

Maximum likelihood estimator

The MLE maximizes the likelihood L(θ) = Π f(x_i; θ), or equivalently the log-likelihood. It is invariant under reparameterization but is not necessarily unbiased.

Method of moments

Equate sample moments to population moments expressed as functions of parameters, then solve for the parameters. The resulting estimator is consistent under mild conditions but may be biased.

Consistency of an estimator

θ̂_n is consistent for θ if θ̂_n converges in probability to θ as n → ∞. Sufficient conditions include asymptotic unbiasedness plus variance tending to zero.

Exponential family

A family with density f(x;θ) = h(x)c(θ)exp(Σ w_j(θ)t_j(x)). It admits sufficient statistics t_j(x) and typically supports complete sufficient statistics and conjugate priors.

Fisher information

I(θ) = E[(∂ log f(X;θ)/∂θ)²] = −E[∂² log f/∂θ²] under regularity. It measures the curvature of the log-likelihood and bounds estimator variance.

Likelihood ratio test

Reject H₀ when Λ = sup_{θ∈Θ₀} L(θ) / sup_{θ∈Θ} L(θ) is small. Under regularity, −2 log Λ is asymptotically chi-square with df equal to the parameter difference.

Central limit theorem

For iid variables with mean μ and finite variance σ², (√n (X̄ − μ))/σ converges in distribution to standard normal N(0,1) as n → ∞, enabling approximate z-based inference.

Bias–variance tradeoff

For an estimator, MSE(θ̂) = Var(θ̂) + [Bias(θ̂)]². Reducing bias often increases variance, so the mean squared error, not bias alone, is the relevant comparison criterion.

Uniformly minimum variance unbiased estimator

A UMVUE is an unbiased estimator whose variance is at most that of every other unbiased estimator for all θ. In exponential families, complete sufficient statistics often yield UMVUEs.