Tamil Nadu HSE Second Year Statistics flashcards

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

Cramer's rule condition

A square system AX = B has a unique solution exactly when det A is nonzero; then each unknown equals the ratio of a determinant with that column replaced by B to det A.

Consistency by rank

The system AX = B is consistent iff rank(A) equals rank of the augmented matrix; if this common rank equals the number of unknowns the solution is unique, otherwise infinitely many.

Area of triangle determinant

The area of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is half the absolute value of the 3x3 determinant having a row of ones and the coordinates.

Nontrivial homogeneous solutions

A homogeneous square system AX = 0 has nonzero solutions iff det A = 0; if det A is nonzero only the trivial solution X = 0 exists.

De Moivre's theorem

For z = r(cos θ + i sin θ) and integer n, z^n equals r^n times cos nθ plus i sin nθ; this gives powers and roots of complex numbers directly.

Cube roots of unity

The three cube roots of 1 are 1, ω = cis(2π/3) and ω²; they satisfy 1 + ω + ω² = 0 and ω³ = 1, so powers of ω repeat with period three.

Modulus and argument

For z = x + iy, modulus |z| = √(x²+y²) and argument θ = tan⁻¹(y/x) adjusted to the correct quadrant; z = r cis θ is the polar form.

Perpendicular bisector locus

The locus of z satisfying |z − z1| = |z − z2| is the perpendicular bisector of the line segment joining the two fixed complex numbers z1 and z2.

Sum of roots of cubic

For ax³ + bx² + cx + d = 0 with roots α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, and αβγ = −d/a.

Remainder theorem

Dividing a polynomial f(x) by (x − a) leaves remainder f(a); consequently (x − a) is a factor of f(x) exactly when f(a) = 0.

Descartes' rule of signs

The number of positive real roots of f(x) is at most the number of sign changes in its coefficients, reduced by even numbers; negative roots use f(−x).

Reciprocal equation substitution

A reciprocal equation has symmetric coefficients; dividing by the middle power and substituting y = x + 1/x lowers the degree by half.

Principal range of tan inverse

The inverse tangent tan⁻¹ x takes values strictly between −π/2 and π/2, which makes it a single-valued function on the whole real line.

Complementary inverse identities

For admissible x: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, and sec⁻¹x + cosec⁻¹x = π/2; these follow from complementary angles.

Tan inverse addition formula

tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy < 1; if xy > 1 the result needs π added or subtracted depending on signs.

Inverse cosine of negative x

cos⁻¹(−x) = π − cos⁻¹x and cot⁻¹(−x) = π − cot⁻¹x, unlike sine and tangent inverses, which are odd functions satisfying f(−x) = −f(x).

Circular permutation formula

The number of distinct circular arrangements of n different objects is (n−1)!; if clockwise and anticlockwise orders count as the same, the number halves.

Combination identity nCr

nCr = n!/(r!(n−r)!) counts unordered selections; it satisfies nCr = nC(n−r) and nCr + nC(r−1) = (n+1)Cr, the Pascal relation.

Gap method for seating

To separate n particular persons among m others, arrange the m first in m! ways, then fill n of the m+1 gaps in (m+1)Pn ways.

Distributing identical objects

The number of ways to distribute r identical objects into n distinct boxes, empty boxes allowed, is (r+n−1)C(n−1), the stars and bars count.

General term of binomial expansion

In (a + b)^n the (r+1)th term is nCr a^(n−r) b^r; setting the power of the variable to zero finds the term independent of it.

Sum of binomial coefficients

Setting both terms to 1 gives Σ nCr = 2^n; setting one term to −1 gives the alternating sum Σ(−1)^r nCr = 0 for n at least 1.

Middle term of binomial

When n is even, the middle term is the (n/2 + 1)th term; when n is odd there are two middle terms, the ((n+1)/2)th and ((n+3)/2)th.

Sum of squares formula

The sum 1² + 2² + … + n² equals n(n+1)(2n+1)/6; the companion sums are Σn = n(n+1)/2 and Σn³ = [n(n+1)/2]².

Infinite geometric series sum

A geometric series with first term a and ratio r converges to a/(1−r) only when |r| < 1; otherwise the sum diverges and no finite value exists.

AM GM HM inequality

For positive numbers, arithmetic mean ≥ geometric mean ≥ harmonic mean, with equality only when all numbers are equal; the harmonic mean of a and b is 2ab/(a+b).

Binomial mean and variance

For X following B(n, p), the mean is np, the variance is npq with q = 1 − p, and the standard deviation is the square root of npq.

Poisson mean equals variance

For a Poisson variable with parameter λ, P(X = r) = e^(−λ)λ^r/r! and both mean and variance equal λ; it models rare events in a fixed interval.

Variance of aX plus b

For constants a and b, E(aX + b) = aE(X) + b while V(aX + b) = a²V(X); adding a constant shifts the distribution without changing spread.

Standard error of mean

The standard error of the sample mean is σ/√n, estimated by s/√n when σ is unknown; it measures how much sample means vary across repeated samples.

Finite population correction

When sampling without replacement from a finite population of size N, the variance of the sample mean is (σ²/n)(N−n)/(N−1), smaller than with replacement.

Central limit theorem

For large sample size n, the sample mean is approximately normally distributed with mean μ and variance σ²/n, whatever the shape of the population distribution.

Unbiased sample variance

The statistic s² = Σ(xi − x̄)²/(n−1) is an unbiased estimator of the population variance σ²; dividing by n instead underestimates it.

Confidence interval for mean

With known σ, a 100(1−α)% interval for μ is x̄ ± z(α/2)·σ/√n; with unknown σ and small n use the t distribution instead.

Type I and Type II errors

A Type I error rejects a true null hypothesis, with probability α; a Type II error fails to reject a false null, with probability β, and power is 1 − β.

Large sample z test for mean

Test statistic z = (x̄ − μ0)/(σ/√n), using s if σ is unknown; reject H0 at level α when |z| exceeds the two-tailed critical value.

Regression coefficient relation

The product of the two regression coefficients b(yx) and b(xy) equals r²; each coefficient equals r times the ratio of the corresponding standard deviations.

Spearman rank correlation

Spearman's coefficient is ρ = 1 − 6Σd²/(n(n²−1)), where d is the difference between ranks of each pair; it measures monotone association.

Regression line through means

The least squares regression line of y on x always passes through the point (x̄, ȳ); its slope is r·(sy/sx) and intercept is ȳ − b·x̄.