Tamil Nadu HSE Second Year Statistics cheat sheet

A system AX = B has a unique solution iff det A ≠ 0; then X = A⁻¹B (Cramer's rule gives xᵢ = det Aᵢ / det A, where Aᵢ replaces column i by B).

If det A = 0 and (adj A)B ≠ 0, the system is inconsistent (no solution); if det A = 0 and (adj A)B = 0, it has infinitely many solutions or none depending on rank consistency.

Cramer's rule applies only to square systems with nonzero determinant; for m ≠ n use rank analysis: consistent iff rank(A) = rank(A|B).

Consistency test: rank(A) = rank(A|B) = number of unknowns → unique solution; rank(A) = rank(A|B) < number of unknowns → infinitely many solutions; rank(A) ≠ rank(A|B) → inconsistent.

A homogeneous system AX = 0 always has the trivial solution; it has nontrivial solutions iff det A = 0 (square case), i.e. rank(A) < n.

Adjoint and inverse: A·adj A = (adj A)·A = |A|I, so A⁻¹ = adj A / |A| when |A| ≠ 0; |adj A| = |A|^(n−1) for an n×n matrix.

Geometric applications: area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) equals ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|; zero determinant means collinear points.

Matrix transformations: rotation by θ is [[cosθ, −sinθ],[sinθ, cosθ]]; reflection in the x-axis is [[1,0],[0,−1]]; a singular transformation matrix collapses area to zero.

Modulus–argument form: z = r(cos θ + i sin θ) with r = |z| = √(x²+y²), θ = arg z = tan⁻¹(y/x) adjusted for quadrant; Euler form z = re^{iθ}.

De Moivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ), valid for integer n; for rational powers it gives one of several roots.

nth roots of z = r cis θ are z_k = r^{1/n} cis((θ + 2kπ)/n), k = 0,1,…,n−1; they lie on a circle of radius r^{1/n} and form a regular n-gon.

Product and quotient: |z₁z₂| = |z₁||z₂|, arg(z₁z₂) = arg z₁ + arg z₂; |z₁/z₂| = |z₁|/|z₂|, arg(z₁/z₂) = arg z₁ − arg z₂.

Conjugate facts: z·z̄ = |z|², z + z̄ = 2Re z, z − z̄ = 2i Im z; z is real iff z = z̄, purely imaginary iff z = −z̄.

Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂| and |z₁ − z₂| ≥ ||z₁| − |z₂||; equality in the first holds when arguments are equal.

Cube roots of unity: 1, ω, ω² with ω = cis(2π/3); 1 + ω + ω² = 0 and ω³ = 1, so ωⁿ cycles with period 3.

Loci: |z − z₀| = r is a circle centre z₀ radius r; |z − z₁| = |z − z₂| is the perpendicular bisector of the segment joining z₁ and z₂.

For ax² + bx + c = 0: roots α, β satisfy α + β = −b/a and αβ = c/a; discriminant Δ = b² − 4ac decides real (Δ ≥ 0) or complex (Δ < 0) roots.

Cubic ax³ + bx² + cx + d = 0: α + β + γ = −b/a, αβ + βγ + γα = c/a, αβγ = −d/a (signs alternate, starting negative).

General rule: in a monic polynomial xⁿ + p₁x^{n−1} + … , the sum of products of roots taken k at a time equals (−1)^k p_k.

Synthetic division by (x − a) gives quotient coefficients and remainder f(a); by the Remainder Theorem f(a) is the remainder, and (x − a) is a factor iff f(a) = 0.

Reciprocal equations have coefficients symmetric (aᵢ = a_{n−i}); substitute y = x + 1/x to reduce degree by half.

To form an equation with roots transformed by y = g(x), eliminate x between y = g(x) and f(x) = 0; e.g. roots doubled means replace x by x/2.

Descartes' rule of signs: the number of positive real roots is at most the number of sign changes in f(x), decreasing by even numbers; negative roots use f(−x).

If a root is repeated, f(x) and f′(x) share that factor; the repeated root is a common root of f and f′.

Principal value ranges: sin⁻¹ x ∈ [−π/2, π/2]; cos⁻¹ x ∈ [0, π]; tan⁻¹ x ∈ (−π/2, π/2); cot⁻¹ x ∈ (0, π); sec⁻¹ x ∈ [0, π]{π/2}; cosec⁻¹ x ∈ [−π/2, π/2]{0}.

Reciprocal identities: sin⁻¹(1/x) = cosec⁻¹ x, cos⁻¹(1/x) = sec⁻¹ x, tan⁻¹(1/x) = cot⁻¹ x for x > 0.

Complementary relations: sin⁻¹ x + cos⁻¹ x = π/2, tan⁻¹ x + cot⁻¹ x = π/2, sec⁻¹ x + cosec⁻¹ x = π/2, valid on the domains.

Negative arguments: sin⁻¹(−x) = −sin⁻¹ x, tan⁻¹(−x) = −tan⁻¹ x, cosec⁻¹(−x) = −cosec⁻¹ x, but cos⁻¹(−x) = π − cos⁻¹ x and cot⁻¹(−x) = π − cot⁻¹ x.

Sum formula: tan⁻¹ x + tan⁻¹ y = tan⁻¹((x+y)/(1−xy)) when xy < 1; add π if xy > 1 and x, y > 0, subtract π if xy > 1 and x, y < 0.

Difference formula: tan⁻¹ x − tan⁻¹ y = tan⁻¹((x−y)/(1+xy)) when xy > −1; adjust by π when xy < −1.

Double-angle: 2 tan⁻¹ x = tan⁻¹(2x/(1−x²)) for |x| < 1; also 2 tan⁻¹ x = sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²)) for x ≥ 0.

Simplification substitutions: for √(a²−x²) put x = a sin θ; for √(a²+x²) put x = a tan θ; for √(x²−a²) put x = a sec θ.

Fundamental principle: if a task splits into independent stages with m and n choices, total ways multiply (mn); mutually exclusive cases add.

nPr = n!/(n−r)! counts ordered selections of r from n distinct objects; nCr = n!/(r!(n−r)!) counts unordered selections.

Key relations: nCr = nC(n−r); nCr + nC(r−1) = (n+1)Cr; nPr = r!·nCr; nC0 = nCn = 1.

Permutations of n objects with p alike of one kind, q alike of another: n!/(p! q! …); circular permutations of n distinct objects = (n−1)!, and necklaces with reversal allowed = (n−1)!/2.

Arrangements with repetition allowed: number of r-permutations from n types is nʳ; number of ways to distribute r identical items into n distinct boxes is (r+n−1)C(n−1).

Number of ways to choose at least one object from n distinct objects is 2ⁿ − 1; total subsets including empty is 2ⁿ.

Gap method: to seat n persons of one kind so that no two are adjacent among m others, arrange the m first ((m)! ways) then place in the m+1 gaps: (m+1)Pn ways.

Total handshakes or lines through n points with no three collinear = nC2; number of diagonals of an n-gon = nC2 − n.

General term: (a + b)ⁿ = Σ nCr a^{n−r} b^{r}; the (r+1)th term is T_{r+1} = nCr a^{n−r} b^{r}.

Number of terms in (a+b)ⁿ is n+1; the middle term is the ((n/2)+1)th when n is even, and there are two middle terms when n is odd.

Coefficients: nCr is greatest for r = n/2 (n even) or r = (n±1)/2 (n odd); binomial coefficients are symmetric, nCr = nC(n−r).

Sum of coefficients: putting a = b = 1 gives Σ nCr = 2ⁿ; putting a = 1, b = −1 gives alternating sum Σ (−1)^r nCr = 0.

Term independent of x: write the general term, collect the power of x, set its exponent to zero, solve for r; r must be a nonnegative integer for the term to exist.

Particular coefficient: equate the exponent of the variable in the general term to the required power and solve for r.

Approximations: (1 + x)ⁿ ≈ 1 + nx for small x; more terms give 1 + nx + n(n−1)x²/2! + … when |x| < 1.

General binomial series for any rational index: (1+x)ⁿ = 1 + nx + n(n−1)x²/2! + … valid for |x| < 1.

Arithmetic progression: tₙ = a + (n−1)d; Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l); the arithmetic mean of a and b is (a+b)/2.

Geometric progression: tₙ = ar^{n−1}; Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1; S∞ = a/(1−r) only when |r| < 1; geometric mean of a and b is √(ab).

Arithmetico-geometric series Σ (a + nd)rⁿ is summed by writing S, rS, subtracting, and using the GP sum on the resulting difference.

Standard finite sums: Σn = n(n+1)/2; Σn² = n(n+1)(2n+1)/6; Σn³ = [n(n+1)/2]².

Method of differences: if tₙ = f(n) − f(n+1), the series telescopes and Sₙ = f(1) − f(n+1).

If a, b, c are in AP then 2b = a + c; in GP b² = ac; in HP the reciprocals 1/a, 1/b, 1/c are in AP.

Harmonic mean of a and b is 2ab/(a+b); for positive numbers AM ≥ GM ≥ HM with equality only when all numbers are equal.

Sum of first n terms of an AP with last term known: Sₙ = (n/2)(first + last); number of terms n = (last − first)/d + 1.

A discrete random variable X has probability mass function p(x) with p(x) ≥ 0 and Σp(x) = 1; the cumulative distribution F(x) = P(X ≤ x) is nondecreasing from 0 to 1.

Expectation E(X) = Σ x·p(x); E(aX + b) = aE(X) + b; E(X²) = Σ x²p(x); variance V(X) = E(X²) − [E(X)]².

Variance property: V(aX + b) = a²V(X); standard deviation σ = √V(X).

Binomial distribution B(n, p): P(X = r) = nCr p^r q^{n−r}, q = 1 − p; mean = np, variance = npq, sd = √(npq).

Binomial conditions: fixed number n of independent Bernoulli trials, constant success probability p, only two outcomes per trial.

Poisson distribution: P(X = r) = e^{−λ}λ^r/r!, r = 0,1,2,…; mean = variance = λ; used for rare events in a fixed interval.

Poisson as a limit of binomial: when n is large and p small with np = λ fixed, B(n, p) tends to Poisson with parameter λ.

For independent variables, E(X + Y) = E(X) + E(Y) and V(X + Y) = V(X) + V(Y); for dependent variables add 2Cov(X, Y).

Simple random sampling: every subset of size n has equal selection probability; with replacement gives independent draws, without replacement gives distinct units.

Systematic sampling selects every kth unit after a random start, with k = N/n; stratified sampling divides the population into homogeneous strata and samples within each.

Cluster sampling selects whole groups; multistage sampling combines stages; quota sampling is non-probability and does not permit valid standard-error calculation.

Sampling distribution of the mean: E(x̄) = μ and V(x̄) = σ²/n for sampling with replacement or from an infinite population.

Finite population correction: V(x̄) = (σ²/n)·(N−n)/(N−1) for sampling without replacement; the factor approaches 1 as N grows.

Standard error of the mean is SE = σ/√n, estimated by s/√n when σ is unknown; quadrupling n halves the standard error.

Central limit theorem: for large n the distribution of x̄ is approximately normal with mean μ and variance σ²/n regardless of the population shape.

Standard error of a proportion is √(pq/n); the sample proportion p̂ is unbiased for the population proportion P.

An estimator is unbiased if E(θ̂) = θ; the sample mean x̄ and sample variance s² = Σ(xᵢ − x̄)²/(n−1) are unbiased for μ and σ².

The divisor n−1 in s² corrects the bias of Σ(xᵢ − x̄)²/n, which underestimates σ².

Consistency: θ̂ is consistent if it converges in probability to θ as n → ∞; x̄ and p̂ are consistent.

Efficiency compares variances among unbiased estimators; the one with smaller variance is more efficient, and the minimum-variance bound is the Cramér–Rao bound.

Sufficiency: a statistic is sufficient if it uses all information in the sample about the parameter; x̄ is sufficient for μ in a normal population.

Interval estimate: x̄ ± z_{α/2}·σ/√n for known σ; with unknown σ and small n use t with n−1 degrees of freedom.

Sample size determination: n = (z_{α/2}σ/E)² for a desired margin of error E in estimating a mean.

For proportions, the confidence interval is p̂ ± z_{α/2}√(p̂q̂/n), valid when np̂ and nq̂ are both at least 5.

Steps: state H₀ and H₁, choose significance level α, select the test statistic, compute it, compare with the critical value, and decide to reject or not reject H₀.

Type I error is rejecting a true H₀ (probability α); Type II error is accepting a false H₀ (probability β); power = 1 − β.

One-tailed test uses H₁: μ > μ₀ or μ < μ₀ with critical value z_α; two-tailed uses H₁: μ ≠ μ₀ with z_{α/2}.

Large-sample mean test: z = (x̄ − μ₀)/(σ/√n), using s when σ is unknown; |z| > z_{α/2} rejects H₀ at level α.

Small-sample mean test with unknown σ: t = (x̄ − μ₀)/(s/√n) with n−1 degrees of freedom; use the t table instead of z.

Proportion test: z = (p̂ − P₀)/√(P₀Q₀/n); difference of means: z = (x̄₁ − x̄₂)/√(σ₁²/n₁ + σ₂²/n₂).

Chi-square goodness of fit: χ² = Σ(O − E)²/E with degrees of freedom (number of categories − 1) or reduced by estimated parameters.

p-value is the probability, under H₀, of a statistic at least as extreme as observed; reject H₀ when p-value < α.

Karl Pearson coefficient: r = Σ(x − x̄)(y − ȳ)/√(Σ(x − x̄)²·Σ(y − ȳ)²); it lies in [−1, 1] and is unit-free.

r = +1 or −1 means perfect linear relation; r = 0 means no linear correlation (not necessarily independence); correlation does not imply causation.

Regression of y on x: ŷ = a + bx with b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² = r·(s_y/s_x) and a = ȳ − b x̄; the line passes through (x̄, ȳ).

Regression of x on y: x̂ = c + dy with d = r·(s_x/s_y); the two regression lines differ unless |r| = 1.

Relation between coefficients: b_yx · b_xy = r², and r = ±√(b_yx·b_xy) with the sign of r matching the sign of the slopes.

Spearman's rank correlation: ρ = 1 − 6Σd²/(n(n²−1)), where d is the difference in ranks; for ties use average ranks.

Coefficient of determination r² is the proportion of variance in y explained by the linear regression on x.

Prediction caution: the regression line is valid only within the observed range of x; extrapolation beyond it is unreliable.