IIT JAM MS cheat sheet

Continuity at a point requires lim(x→a) f(x) = f(a); differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| at 0).

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0 for some c in (a,b); Lagrange's MVT drops the f(a)=f(b) condition and gives f'(c) = (f(b)−f(a))/(b−a).

L'Hôpital applies only to 0/0 or ∞/∞ indeterminate forms; verify the form before differentiating numerator and denominator separately.

Taylor expansion: f(x) = Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + R_n(x); for convergence tests compare |x−a| with the radius from the ratio test on coefficients.

A real sequence converges iff it is Cauchy; a bounded monotone sequence always converges. For series, Σa_n converges absolutely implies convergence, but conditional convergence allows rearrangement to change the sum.

For a 2×2 matrix A, det(A)=ad−bc, trace = a+d, eigenvalues solve λ² − (tr A)λ + det A = 0; A is invertible iff det A ≠ 0 and diagonalizable over R iff it has 2 independent eigenvectors.

Rank–nullity: rank(T) + nullity(T) = dim(domain). A set of n vectors in R^n is a basis iff its determinant is nonzero; linear independence alone is insufficient in higher dimension.

Group theory: Lagrange's theorem gives |H| divides |G| for a subgroup H; the order of an element divides |G|. A group of prime order is cyclic, and every group of order p² is abelian.

Integration: ∫₀^{π/2} sin^n x dx = ∫₀^{π/2} cos^n x dx follows the Wallis recurrence I_n = ((n−1)/n) I_{n−2} with I_0 = π/2 and I_1 = 1.

Estimator bias: E(θ̂) − θ = bias. Sample variance S² = Σ(X_i − X̄)²/(n−1) is unbiased for σ², while the MLE σ̂² = Σ(X_i − X̄)²/n is biased by factor (n−1)/n.

Cramér–Rao bound: Var(θ̂) ≥ 1/(n I(θ)), where I(θ) = E[(∂ log f/∂θ)²]. An unbiased estimator attaining the bound is the UMVUE in the regular exponential-family setting.

Sufficiency: by the factorization theorem, T is sufficient iff the joint density factors as g(T;θ)h(x). In a one-parameter exponential family, the natural statistic is complete and sufficient.

Confidence interval for a normal mean with unknown variance uses t_{n−1}: X̄ ± t_{α/2,n−1} · S/√n. Replace t by z only when σ is known or n is large.

Hypothesis testing: the p-value is P(reject-region statistic at least as extreme | H₀ true); reject at level α if p ≤ α. Type I error = α, Type II error = β, power = 1 − β.

Neyman–Pearson lemma: the most powerful test of simple H₀ vs simple H₁ rejects for large values of the likelihood ratio f₁(x)/f₀(x); the threshold is set by the size α.

Bayes: posterior ∝ prior × likelihood. For a Beta(a,b) prior with Binomial(n,θ) data, the posterior is Beta(a + Σx_i, b + n − Σx_i), so the prior is conjugate.

Distribution facts: if X_i ~ N(μ,σ²) iid, then X̄ ~ N(μ, σ²/n) and (n−1)S²/σ² ~ χ²_{n−1}, independent of X̄; the ratio of independent χ²/df gives an F distribution.

Continuity at a point requires lim(x→a) f(x) = f(a); differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| at 0).

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0 for some c in (a,b); Lagrange's MVT drops the f(a)=f(b) condition and gives f'(c) = (f(b)−f(a))/(b−a).

L'Hôpital applies only to 0/0 or ∞/∞ indeterminate forms; verify the form before differentiating numerator and denominator separately.

Taylor expansion: f(x) = Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + R_n(x); for convergence tests compare |x−a| with the radius from the ratio test on coefficients.

A real sequence converges iff it is Cauchy; a bounded monotone sequence always converges. For series, Σa_n converges absolutely implies convergence, but conditional convergence allows rearrangement to change the sum.

For a 2×2 matrix A, det(A)=ad−bc, trace = a+d, eigenvalues solve λ² − (tr A)λ + det A = 0; A is invertible iff det A ≠ 0 and diagonalizable over R iff it has 2 independent eigenvectors.

Rank–nullity: rank(T) + nullity(T) = dim(domain). A set of n vectors in R^n is a basis iff its determinant is nonzero; linear independence alone is insufficient in higher dimension.

Group theory: Lagrange's theorem gives |H| divides |G| for a subgroup H; the order of an element divides |G|. A group of prime order is cyclic, and every group of order p² is abelian.

Integration: ∫₀^{π/2} sin^n x dx = ∫₀^{π/2} cos^n x dx follows the Wallis recurrence I_n = ((n−1)/n) I_{n−2} with I_0 = π/2 and I_1 = 1.

Estimator bias: E(θ̂) − θ = bias. Sample variance S² = Σ(X_i − X̄)²/(n−1) is unbiased for σ², while the MLE σ̂² = Σ(X_i − X̄)²/n is biased by factor (n−1)/n.

Cramér–Rao bound: Var(θ̂) ≥ 1/(n I(θ)), where I(θ) = E[(∂ log f/∂θ)²]. An unbiased estimator attaining the bound is the UMVUE in the regular exponential-family setting.

Sufficiency: by the factorization theorem, T is sufficient iff the joint density factors as g(T;θ)h(x). In a one-parameter exponential family, the natural statistic is complete and sufficient.

Confidence interval for a normal mean with unknown variance uses t_{n−1}: X̄ ± t_{α/2,n−1} · S/√n. Replace t by z only when σ is known or n is large.

Hypothesis testing: the p-value is P(reject-region statistic at least as extreme | H₀ true); reject at level α if p ≤ α. Type I error = α, Type II error = β, power = 1 − β.

Neyman–Pearson lemma: the most powerful test of simple H₀ vs simple H₁ rejects for large values of the likelihood ratio f₁(x)/f₀(x); the threshold is set by the size α.

Bayes: posterior ∝ prior × likelihood. For a Beta(a,b) prior with Binomial(n,θ) data, the posterior is Beta(a + Σx_i, b + n − Σx_i), so the prior is conjugate.

Distribution facts: if X_i ~ N(μ,σ²) iid, then X̄ ~ N(μ, σ²/n) and (n−1)S²/σ² ~ χ²_{n−1}, independent of X̄; the ratio of independent χ²/df gives an F distribution.

Continuity at a point requires lim(x→a) f(x) = f(a); differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| at 0).

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0 for some c in (a,b); Lagrange's MVT drops the f(a)=f(b) condition and gives f'(c) = (f(b)−f(a))/(b−a).

L'Hôpital applies only to 0/0 or ∞/∞ indeterminate forms; verify the form before differentiating numerator and denominator separately.

Taylor expansion: f(x) = Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + R_n(x); for convergence tests compare |x−a| with the radius from the ratio test on coefficients.

A real sequence converges iff it is Cauchy; a bounded monotone sequence always converges. For series, Σa_n converges absolutely implies convergence, but conditional convergence allows rearrangement to change the sum.

For a 2×2 matrix A, det(A)=ad−bc, trace = a+d, eigenvalues solve λ² − (tr A)λ + det A = 0; A is invertible iff det A ≠ 0 and diagonalizable over R iff it has 2 independent eigenvectors.

Rank–nullity: rank(T) + nullity(T) = dim(domain). A set of n vectors in R^n is a basis iff its determinant is nonzero; linear independence alone is insufficient in higher dimension.

Group theory: Lagrange's theorem gives |H| divides |G| for a subgroup H; the order of an element divides |G|. A group of prime order is cyclic, and every group of order p² is abelian.

Integration: ∫₀^{π/2} sin^n x dx = ∫₀^{π/2} cos^n x dx follows the Wallis recurrence I_n = ((n−1)/n) I_{n−2} with I_0 = π/2 and I_1 = 1.

Estimator bias: E(θ̂) − θ = bias. Sample variance S² = Σ(X_i − X̄)²/(n−1) is unbiased for σ², while the MLE σ̂² = Σ(X_i − X̄)²/n is biased by factor (n−1)/n.

Cramér–Rao bound: Var(θ̂) ≥ 1/(n I(θ)), where I(θ) = E[(∂ log f/∂θ)²]. An unbiased estimator attaining the bound is the UMVUE in the regular exponential-family setting.

Sufficiency: by the factorization theorem, T is sufficient iff the joint density factors as g(T;θ)h(x). In a one-parameter exponential family, the natural statistic is complete and sufficient.

Confidence interval for a normal mean with unknown variance uses t_{n−1}: X̄ ± t_{α/2,n−1} · S/√n. Replace t by z only when σ is known or n is large.

Hypothesis testing: the p-value is P(reject-region statistic at least as extreme | H₀ true); reject at level α if p ≤ α. Type I error = α, Type II error = β, power = 1 − β.

Neyman–Pearson lemma: the most powerful test of simple H₀ vs simple H₁ rejects for large values of the likelihood ratio f₁(x)/f₀(x); the threshold is set by the size α.

Bayes: posterior ∝ prior × likelihood. For a Beta(a,b) prior with Binomial(n,θ) data, the posterior is Beta(a + Σx_i, b + n − Σx_i), so the prior is conjugate.

Distribution facts: if X_i ~ N(μ,σ²) iid, then X̄ ~ N(μ, σ²/n) and (n−1)S²/σ² ~ χ²_{n−1}, independent of X̄; the ratio of independent χ²/df gives an F distribution.

Continuity at a point requires lim(x→a) f(x) = f(a); differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| at 0).

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0 for some c in (a,b); Lagrange's MVT drops the f(a)=f(b) condition and gives f'(c) = (f(b)−f(a))/(b−a).

L'Hôpital applies only to 0/0 or ∞/∞ indeterminate forms; verify the form before differentiating numerator and denominator separately.

Taylor expansion: f(x) = Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + R_n(x); for convergence tests compare |x−a| with the radius from the ratio test on coefficients.

A real sequence converges iff it is Cauchy; a bounded monotone sequence always converges. For series, Σa_n converges absolutely implies convergence, but conditional convergence allows rearrangement to change the sum.

For a 2×2 matrix A, det(A)=ad−bc, trace = a+d, eigenvalues solve λ² − (tr A)λ + det A = 0; A is invertible iff det A ≠ 0 and diagonalizable over R iff it has 2 independent eigenvectors.

Rank–nullity: rank(T) + nullity(T) = dim(domain). A set of n vectors in R^n is a basis iff its determinant is nonzero; linear independence alone is insufficient in higher dimension.

Group theory: Lagrange's theorem gives |H| divides |G| for a subgroup H; the order of an element divides |G|. A group of prime order is cyclic, and every group of order p² is abelian.

Integration: ∫₀^{π/2} sin^n x dx = ∫₀^{π/2} cos^n x dx follows the Wallis recurrence I_n = ((n−1)/n) I_{n−2} with I_0 = π/2 and I_1 = 1.

Estimator bias: E(θ̂) − θ = bias. Sample variance S² = Σ(X_i − X̄)²/(n−1) is unbiased for σ², while the MLE σ̂² = Σ(X_i − X̄)²/n is biased by factor (n−1)/n.

Cramér–Rao bound: Var(θ̂) ≥ 1/(n I(θ)), where I(θ) = E[(∂ log f/∂θ)²]. An unbiased estimator attaining the bound is the UMVUE in the regular exponential-family setting.

Sufficiency: by the factorization theorem, T is sufficient iff the joint density factors as g(T;θ)h(x). In a one-parameter exponential family, the natural statistic is complete and sufficient.

Confidence interval for a normal mean with unknown variance uses t_{n−1}: X̄ ± t_{α/2,n−1} · S/√n. Replace t by z only when σ is known or n is large.

Hypothesis testing: the p-value is P(reject-region statistic at least as extreme | H₀ true); reject at level α if p ≤ α. Type I error = α, Type II error = β, power = 1 − β.

Neyman–Pearson lemma: the most powerful test of simple H₀ vs simple H₁ rejects for large values of the likelihood ratio f₁(x)/f₀(x); the threshold is set by the size α.

Bayes: posterior ∝ prior × likelihood. For a Beta(a,b) prior with Binomial(n,θ) data, the posterior is Beta(a + Σx_i, b + n − Σx_i), so the prior is conjugate.

Distribution facts: if X_i ~ N(μ,σ²) iid, then X̄ ~ N(μ, σ²/n) and (n−1)S²/σ² ~ χ²_{n−1}, independent of X̄; the ratio of independent χ²/df gives an F distribution.

Continuity at a point requires lim(x→a) f(x) = f(a); differentiability implies continuity, but continuity does not imply differentiability (e.g., |x| at 0).

Rolle's theorem: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then f'(c)=0 for some c in (a,b); Lagrange's MVT drops the f(a)=f(b) condition and gives f'(c) = (f(b)−f(a))/(b−a).

L'Hôpital applies only to 0/0 or ∞/∞ indeterminate forms; verify the form before differentiating numerator and denominator separately.

Taylor expansion: f(x) = Σ_{k=0}^{n} f^(k)(a)(x−a)^k/k! + R_n(x); for convergence tests compare |x−a| with the radius from the ratio test on coefficients.

A real sequence converges iff it is Cauchy; a bounded monotone sequence always converges. For series, Σa_n converges absolutely implies convergence, but conditional convergence allows rearrangement to change the sum.

For a 2×2 matrix A, det(A)=ad−bc, trace = a+d, eigenvalues solve λ² − (tr A)λ + det A = 0; A is invertible iff det A ≠ 0 and diagonalizable over R iff it has 2 independent eigenvectors.

Rank–nullity: rank(T) + nullity(T) = dim(domain). A set of n vectors in R^n is a basis iff its determinant is nonzero; linear independence alone is insufficient in higher dimension.

Group theory: Lagrange's theorem gives |H| divides |G| for a subgroup H; the order of an element divides |G|. A group of prime order is cyclic, and every group of order p² is abelian.

Integration: ∫₀^{π/2} sin^n x dx = ∫₀^{π/2} cos^n x dx follows the Wallis recurrence I_n = ((n−1)/n) I_{n−2} with I_0 = π/2 and I_1 = 1.

Estimator bias: E(θ̂) − θ = bias. Sample variance S² = Σ(X_i − X̄)²/(n−1) is unbiased for σ², while the MLE σ̂² = Σ(X_i − X̄)²/n is biased by factor (n−1)/n.

Cramér–Rao bound: Var(θ̂) ≥ 1/(n I(θ)), where I(θ) = E[(∂ log f/∂θ)²]. An unbiased estimator attaining the bound is the UMVUE in the regular exponential-family setting.

Sufficiency: by the factorization theorem, T is sufficient iff the joint density factors as g(T;θ)h(x). In a one-parameter exponential family, the natural statistic is complete and sufficient.

Confidence interval for a normal mean with unknown variance uses t_{n−1}: X̄ ± t_{α/2,n−1} · S/√n. Replace t by z only when σ is known or n is large.

Hypothesis testing: the p-value is P(reject-region statistic at least as extreme | H₀ true); reject at level α if p ≤ α. Type I error = α, Type II error = β, power = 1 − β.

Neyman–Pearson lemma: the most powerful test of simple H₀ vs simple H₁ rejects for large values of the likelihood ratio f₁(x)/f₀(x); the threshold is set by the size α.

Bayes: posterior ∝ prior × likelihood. For a Beta(a,b) prior with Binomial(n,θ) data, the posterior is Beta(a + Σx_i, b + n − Σx_i), so the prior is conjugate.

Distribution facts: if X_i ~ N(μ,σ²) iid, then X̄ ~ N(μ, σ²/n) and (n−1)S²/σ² ~ χ²_{n−1}, independent of X̄; the ratio of independent χ²/df gives an F distribution.