# Trigonometry, Sequences, Probability & Statistics > Mathematical foundations for machine learning: 32 concept-review cards on trigonometry, complex-number representations, sequences and series, combinatorics, conditional probability, Bayes' theorem, distributions, and hypothesis testing. - **Canonical URL:** https://flashcards.world/flashcards/sets/5ffbff64-fe75-482e-aa9f-c511ed096fa8/ - **Markdown URL:** https://flashcards.world/flashcards/sets/5ffbff64-fe75-482e-aa9f-c511ed096fa8.md - **Level:** intermediate - **Languages:** en → en - **Total Cards:** 32 ## Flashcards (32 Cards) | # | Front (Term / Prompt) | Back (Definition / Answer) | Example | Multiple-Choice Options | | :--- | :--- | :--- | :--- | :--- | | 1 | For a geometric sequence with first term a₁ and nonzero ratio r ≠ 1, what are a_n and the sum S_n of its first n terms? | For integers n ≥ 1: a_n = a₁r^(n − 1), and S_n = a₁(1 − r^n)/(1 − r). | | | | 2 | For a normal distribution, approximately what proportions lie within 1, 2 and 3 standard deviations of its mean? | About 68.27%, 95.45% and 99.73%, respectively. These are distribution probabilities, not guaranteed percentages in each finite sample. | | | | 3 | What is a p-value, and how is it used with a preselected significance level α? | Under H₀ and the test’s assumptions, it is the probability of a test statistic at least as extreme as observed, in the direction(s) specified by the test. Reject H₀ when p ≤ α. It is not the probability that H₀ is true. | | | | 4 | 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]. | | | | 5 | For an arithmetic sequence with first term a₁ and common difference d, what are a_n and the sum S_n of its first n terms? | For integers n ≥ 1: a_n = a₁ + (n − 1)d, and S_n = n(a₁ + a_n)/2 = n[2a₁ + (n − 1)d]/2. | | | | 6 | For y = A sin(Bx − C), with A ≠ 0 and B > 0, what are the amplitude, period and phase shift? | Amplitude: \|A\|. Period: 2π/B. Phase shift: C/B, to the right when positive and to the left when negative. | | | | 7 | For an ellipse centered at (h, k) with horizontal semimajor axis a > b > 0, what are its equation and focal distance? | (x − h)²/a² + (y − k)²/b² = 1. The distance c from the center to either focus satisfies c² = a² − b². | | | | 8 | What is statistical power for a specified alternative hypothesis? | Power = 1 − β: the probability of rejecting H₀ when that specified alternative is true. | | | | 9 | What does De Moivre’s theorem give for z = r(cos θ + i sin θ), with r > 0 and integer n? | z^n = r^n[cos(nθ) + i sin(nθ)] = r^n e^(inθ). | | | | 10 | 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). | | | | 11 | How is a z-score calculated, and does standardizing a variable make it normally distributed? | For mean μ and standard deviation σ > 0, z = (x − μ)/σ gives the signed distance from the mean in standard deviations. Standardization alone does not create normality; Z is standard normal if X is normal. | | | | 12 | 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)!). | | | | 13 | How do you calculate the finite expected value of a discrete random variable X? | E[X] = Σ xP(X = x), summed over its possible values. For infinitely many values, a finite expectation requires Σ \|x\|P(X = x) < ∞. | | | | 14 | 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. | | | | 15 | State the Addition Rule of Probability for any two events A and B. | P(A ∪ B) = P(A) + P(B) - P(A ∩ B). | | | | 16 | 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). | | | | 17 | For a random variable X with finite second moment, how can you compute its variance from moments? | Var(X) = E[X²] − (E[X])² = σ². | | | | 18 | How do you convert between polar and Cartesian coordinates, choosing r ≥ 0? | x = r cos θ and y = r sin θ. Conversely, r = √(x² + y²); away from the origin, θ = atan2(y, x), up to multiples of 2π. At the origin, r = 0 and θ is arbitrary. | | | | 19 | How is radian measure defined, and how do you convert a degree measure to radians? | One radian subtends an arc whose length equals the radius. Multiply the numerical degree measure by π/180 to obtain the radian measure. | | | | 20 | For n ≥ 2 independent, identically distributed observations with finite variance σ², why does sample variance divide by n − 1? | s² = Σ(x_i − x̄)²/(n − 1) is unbiased for σ². Estimating the mean from the same data makes E[Σ(X_i − X̄)²] = (n − 1)σ². | | | | 21 | When selecting k distinct objects from n distinct objects without replacement, how do permutations and combinations differ? | For integers 0 ≤ k ≤ n, ordered selections number P(n, k) = n!/(n − k)!; unordered selections number C(n, k) = n!/[k!(n − k)!]. | | | | 22 | On the unit circle, how are cos θ and sin θ related to a point reached by an angle θ in standard position? | The point has coordinates (cos θ, sin θ): cosine is its x-coordinate and sine its y-coordinate. | | | | 23 | What condition defines independence of two events A and B? | P(A ∩ B) = P(A)P(B). If P(B) > 0, this is equivalent to P(A \| B) = P(A). | | | | 24 | What are the Pythagorean trigonometric identity and its tangent/secant and cotangent/cosecant forms? | sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ when cos θ ≠ 0. Dividing by sin²θ gives 1 + cot²θ = csc²θ when sin θ ≠ 0. | | | | 25 | 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). | | | | 26 | What does the classical central limit theorem say about the mean of independent, identically distributed observations? | With finite mean μ and variance 0 < σ² < ∞, √n(X̄ − μ)/σ converges in distribution to N(0, 1) as n grows. Thus X̄ may be approximated by N(μ, σ²/n); accuracy depends on the distribution and n, with no universal n ≥ 30 guarantee. | | | | 27 | Under independent, identically distributed sampling with finite variance σ², what is the standard error of the sample mean? | SEM = σ/√n. For fixed σ, it decreases in proportion to 1/√n. When σ is unknown, s/√n estimates the SEM using the sample standard deviation s (n ≥ 2). | | | | 28 | What is the difference between Type I and Type II errors in hypothesis testing? | Type I: rejecting a true null hypothesis H₀. Type II: failing to reject H₀ under a specified alternative where it is false. The significance level α controls Type I error; β denotes the Type II error probability at that alternative. | | | | 29 | For radius r > 0 and central angle 0 ≤ θ ≤ 2π in radians, what are the arc length and sector area? | Arc length: s = rθ. Sector area: A = r²θ/2. | | | | 30 | For n independent trials with the same success probability 0 < p < 1, what are the binomial success-count probability, mean and variance? | For integers n ≥ 1 and k = 0, …, n: P(X = k) = C(n, k)p^k(1 − p)^(n − k). E[X] = np; Var(X) = np(1 − p). | | | | 31 | If 0 < P(A) < 1 and P(B) > 0, how does Bayes’ theorem express P(A \| B) using A and its complement Aᶜ? | P(A \| B) = P(B \| A)P(A)/P(B), where P(B) = P(B \| A)P(A) + P(B \| Aᶜ)P(Aᶜ). | | | | 32 | When does the infinite geometric series a₁ + a₁r + a₁r² + … converge, and what is its sum? | If a₁ ≠ 0, it converges exactly when \|r\| < 1, with sum a₁/(1 − r). If a₁ = 0, every term is zero and the sum is 0 for any real r. | | | --- Study this deck interactively with FSRS-6 spaced repetition, audio review, and multiple-choice mode at [Flashcards World](https://flashcards.world/flashcards/sets/5ffbff64-fe75-482e-aa9f-c511ed096fa8/).