Why is (n - 1) used as the divisor instead of n in the sample variance formula s^2 = Σ(x_i - x_bar)^2 / (n - 1)?

Bessel's correction: dividing by (n - 1) corrects downward bias, providing an unbiased estimator of population variance σ^2.

State the Binomial Theorem for expanding (x + y)^n for a positive integer n.

(x + y)^n = Σ [k=0 to n] (n choose k) * x^(n - k) * y^k, where (n choose k) = n! / (k! * (n - k)!).

What is the computational formula for the variance Var(X) of a random variable X?

Var(X) = σ^2 = E[X^2] - (E[X])^2.

Under what condition does an infinite geometric series converge, and what is its sum formula?

Converges if and only if |r| < 1; sum formula is S_∞ = a_1 / (1 - r).

What are the standard equation of an ellipse centered at (h, k) and the relationship between semi-axes a, b and focal distance c?

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1; focal relationship is c^2 = a^2 - b^2 (assuming horizontal major axis a > b).

How is radian measure defined, and what conversion factor converts degrees to radians?

One radian is the central angle subtending an arc length equal to the circle's radius (s = r); conversion factor is θ_rad = θ_deg * (π / 180°).

What are the probability mass function, mean, and variance of a Binomial distribution B(n, p)?

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k); Mean μ = n*p; Variance σ^2 = n*p*(1 - p).

How is a data point x converted to a standard normal z-score, and what does the z-score represent?

z = (x - μ) / σ; it represents the signed number of standard deviations the value x lies above or below the mean.

On the unit circle (radius r = 1), how are cos(θ) and sin(θ) defined in terms of Cartesian coordinates?

cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the terminal point on the circle: (x, y) = (cos θ, sin θ).

How is the expected value E[X] of a discrete random variable X calculated?

E[X] = μ = Σ [x_i * P(X = x_i)] over all possible values x_i.

State the Law of Sines and the Law of Cosines for a triangle with sides a, b, c and opposite angles A, B, C.

Law of Sines: a / sin(A) = b / sin(B) = c / sin(C); Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C).

What are the period, amplitude, and horizontal phase shift of y = A*sin(Bx - C) with B > 0?

Amplitude = |A|; Period T = 2π / B; Phase shift = C / B (to the right if C/B > 0).

What is statistical power, and how is it related to the Type II error rate (β)?

Power = 1 - β; it is the probability of correctly rejecting a false null hypothesis.

What are the trigonometric addition formulas for sin(α + β) and cos(α + β)?

sin(α + β) = sin(α)cos(β) + cos(α)sin(β); cos(α + β) = cos(α)cos(β) - sin(α)sin(β).

State the fundamental Pythagorean trigonometric identity and its two derived identities involving tan(θ) and cot(θ).

1. sin^2(θ) + cos^2(θ) = 1 2. 1 + tan^2(θ) = sec^2(θ) 3. 1 + cot^2(θ) = csc^2(θ).

Distinguish between a Type I error (α) and a Type II error (β) in statistical hypothesis testing.

Type I error (false positive) rejects the null hypothesis H0 when H0 is true; Type II error (false negative) fails to reject H0 when H0 is false.

What are the domains and principal ranges of the inverse trigonometric functions arcsin(x) and arccos(x)?

arcsin(x): domain [-1, 1], range [-π/2, π/2]; arccos(x): domain [-1, 1], range [0, π].

What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of a geometric sequence with common ratio r != 1?

a_n = a_1 * r^(n - 1); S_n = a_1 * (1 - r^n) / (1 - r).

State Bayes' Theorem for revising the probability of hypothesis A given observed evidence B.

P(A | B) = [P(B | A) * P(A)] / P(B) = [P(B | A) * P(A)] / [P(B | A)*P(A) + P(B | ¬A)*P(¬A)].

What mathematical test establishes that two events A and B are statistically independent?

P(A ∩ B) = P(A) * P(B), or equivalently P(A | B) = P(A).

State the Addition Rule of Probability for any two events A and B.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B).

What does the Central Limit Theorem (CLT) state regarding sample means from any population distribution?

For independent, identically distributed samples of sufficiently large size (n >= 30), the sample mean X_bar is approximately normally distributed: X_bar ~ N(μ, σ^2 / n).

What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of an arithmetic sequence?

a_n = a_1 + (n - 1)*d; S_n = (n / 2) * (a_1 + a_n) = (n / 2) * [2*a_1 + (n - 1)*d].

What is the standard error of the mean (SEM), and how does increasing sample size n affect it?

SEM = σ / √n; increasing sample size decreases standard error inversely proportional to the square root of n, increasing estimate precision.

How do permutations and combinations differ in combinatorics, and what are their computational formulas?

Permutations account for order: P(n, k) = n! / (n - k)!; Combinations ignore order: C(n, k) = (n choose k) = n! / (k! * (n - k)!).

What are the formulas for arc length (s) and area of a circular sector (A) subtended by central angle θ in radians?

Arc length: s = r * θ; Sector area: A = (1/2) * r^2 * θ.

State the double-angle formulas for sin(2θ) and the three standard forms for cos(2θ).

sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos^2(θ) - sin^2(θ) = 2cos^2(θ) - 1 = 1 - 2sin^2(θ).

State De Moivre's Theorem for raising a complex number in polar form to the integer power n.

[r*(cos θ + i*sin θ)]^n = r^n * [cos(nθ) + i*sin(nθ)] = r^n * e^(i*n*θ).

What percentages of data fall within 1, 2, and 3 standard deviations of the mean in a Normal distribution (Empirical Rule)?

Approximately 68.27% within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ.

How do you convert polar coordinates (r, θ) to Cartesian coordinates (x, y) and vice versa?

Polar to Cartesian: x = r*cos(θ), y = r*sin(θ); Cartesian to polar: r = √(x^2 + y^2), θ = atan2(y, x).

What is the statistical definition and practical interpretation of a p-value?

The probability of obtaining test results at least as extreme as observed, assuming the null hypothesis H0 is true; if p <= α, reject H0.

What is the formula for the conditional probability of event A given event B has occurred, P(A | B)?

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

Trigonometry (Italiano) — Intermedio (B1/B2) 2: Trigonometry, Precalculus, Probability & Statistics

[Italiano] Trigonometry (Italiano): Unit circle, trigonometric identities, limits, sequences, permutations, combinations, normal distribution, and hypothesis testing basics

Level: intermediate
Languages: it → it
Cards: 32
Markdown: Trigonometry (Italiano) — Intermedio (B1/B2) 2: Trigonometry, Precalculus, Probability & Statistics (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	Why is (n - 1) used as the divisor instead of n in the sample variance formula s^2 = Σ(x_i - x_bar)^2 / (n - 1)?	Bessel's correction: dividing by (n - 1) corrects downward bias, providing an unbiased estimator of population variance σ^2.
2	State the Binomial Theorem for expanding (x + y)^n for a positive integer n.	(x + y)^n = Σ [k=0 to n] (n choose k) * x^(n - k) * y^k, where (n choose k) = n! / (k! * (n - k)!).
3	What is the computational formula for the variance Var(X) of a random variable X?	Var(X) = σ^2 = E[X^2] - (E[X])^2.
4	Under what condition does an infinite geometric series converge, and what is its sum formula?	Converges if and only if \|r\| < 1; sum formula is S_∞ = a_1 / (1 - r).
5	What are the standard equation of an ellipse centered at (h, k) and the relationship between semi-axes a, b and focal distance c?	(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1; focal relationship is c^2 = a^2 - b^2 (assuming horizontal major axis a > b).
6	How is radian measure defined, and what conversion factor converts degrees to radians?	One radian is the central angle subtending an arc length equal to the circle's radius (s = r); conversion factor is θ_rad = θ_deg * (π / 180°).
7	What are the probability mass function, mean, and variance of a Binomial distribution B(n, p)?	P(X = k) = (n choose k) * p^k * (1 - p)^(n - k); Mean μ = np; Variance σ^2 = np*(1 - p).
8	How is a data point x converted to a standard normal z-score, and what does the z-score represent?	z = (x - μ) / σ; it represents the signed number of standard deviations the value x lies above or below the mean.
9	On the unit circle (radius r = 1), how are cos(θ) and sin(θ) defined in terms of Cartesian coordinates?	cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the terminal point on the circle: (x, y) = (cos θ, sin θ).
10	How is the expected value E[X] of a discrete random variable X calculated?	E[X] = μ = Σ [x_i * P(X = x_i)] over all possible values x_i.
11	State the Law of Sines and the Law of Cosines for a triangle with sides a, b, c and opposite angles A, B, C.	Law of Sines: a / sin(A) = b / sin(B) = c / sin(C); Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C).
12	What are the period, amplitude, and horizontal phase shift of y = A*sin(Bx - C) with B > 0?	Amplitude = \|A\|; Period T = 2π / B; Phase shift = C / B (to the right if C/B > 0).
13	What is statistical power, and how is it related to the Type II error rate (β)?	Power = 1 - β; it is the probability of correctly rejecting a false null hypothesis.
14	What are the trigonometric addition formulas for sin(α + β) and cos(α + β)?	sin(α + β) = sin(α)cos(β) + cos(α)sin(β); cos(α + β) = cos(α)cos(β) - sin(α)sin(β).
15	State the fundamental Pythagorean trigonometric identity and its two derived identities involving tan(θ) and cot(θ).	1. sin^2(θ) + cos^2(θ) = 1 2. 1 + tan^2(θ) = sec^2(θ) 3. 1 + cot^2(θ) = csc^2(θ).
16	Distinguish between a Type I error (α) and a Type II error (β) in statistical hypothesis testing.	Type I error (false positive) rejects the null hypothesis H0 when H0 is true; Type II error (false negative) fails to reject H0 when H0 is false.
17	What are the domains and principal ranges of the inverse trigonometric functions arcsin(x) and arccos(x)?	arcsin(x): domain [-1, 1], range [-π/2, π/2]; arccos(x): domain [-1, 1], range [0, π].
18	What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of a geometric sequence with common ratio r != 1?	a_n = a_1 * r^(n - 1); S_n = a_1 * (1 - r^n) / (1 - r).
19	State Bayes' Theorem for revising the probability of hypothesis A given observed evidence B.	P(A \| B) = [P(B \| A) * P(A)] / P(B) = [P(B \| A) * P(A)] / [P(B \| A)P(A) + P(B \| ¬A)P(¬A)].
20	What mathematical test establishes that two events A and B are statistically independent?	P(A ∩ B) = P(A) * P(B), or equivalently P(A \| B) = P(A).
21	State the Addition Rule of Probability for any two events A and B.	P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
22	What does the Central Limit Theorem (CLT) state regarding sample means from any population distribution?	For independent, identically distributed samples of sufficiently large size (n >= 30), the sample mean X_bar is approximately normally distributed: X_bar ~ N(μ, σ^2 / n).
23	What are the formulas for the n-th term (a_n) and sum of first n terms (S_n) of an arithmetic sequence?	a_n = a_1 + (n - 1)d; S_n = (n / 2) (a_1 + a_n) = (n / 2) * [2a_1 + (n - 1)d].
24	What is the standard error of the mean (SEM), and how does increasing sample size n affect it?	SEM = σ / √n; increasing sample size decreases standard error inversely proportional to the square root of n, increasing estimate precision.
25	How do permutations and combinations differ in combinatorics, and what are their computational formulas?	Permutations account for order: P(n, k) = n! / (n - k)!; Combinations ignore order: C(n, k) = (n choose k) = n! / (k! * (n - k)!).
26	What are the formulas for arc length (s) and area of a circular sector (A) subtended by central angle θ in radians?	Arc length: s = r * θ; Sector area: A = (1/2) * r^2 * θ.
27	State the double-angle formulas for sin(2θ) and the three standard forms for cos(2θ).	sin(2θ) = 2sin(θ)cos(θ); cos(2θ) = cos^2(θ) - sin^2(θ) = 2cos^2(θ) - 1 = 1 - 2sin^2(θ).
28	State De Moivre's Theorem for raising a complex number in polar form to the integer power n.	[r(cos θ + isin θ)]^n = r^n * [cos(nθ) + isin(nθ)] = r^n e^(inθ).
29	What percentages of data fall within 1, 2, and 3 standard deviations of the mean in a Normal distribution (Empirical Rule)?	Approximately 68.27% within μ ± 1σ, 95.45% within μ ± 2σ, and 99.73% within μ ± 3σ.
30	How do you convert polar coordinates (r, θ) to Cartesian coordinates (x, y) and vice versa?	Polar to Cartesian: x = rcos(θ), y = rsin(θ); Cartesian to polar: r = √(x^2 + y^2), θ = atan2(y, x).
31	What is the statistical definition and practical interpretation of a p-value?	The probability of obtaining test results at least as extreme as observed, assuming the null hypothesis H0 is true; if p <= α, reject H0.
32	What is the formula for the conditional probability of event A given event B has occurred, P(A \| B)?	P(A \| B) = P(A ∩ B) / P(B), provided P(B) > 0.