概率模型、极限、积分、矩阵和微分方程

• This document provides study notes on classical probability models, geometric probability models, limits, series, the properties of definite integrals, determinants, matrices, linear spaces, and differential equations.

Classical vs. Geometric Probability Models

• Basic event characteristics: mutually exclusive; any event (except impossible ones) can be expressed as the sum of basic events.
• Classical probability model: finite, mutually exclusive outcomes.
• If there are n equally likely outcomes, each basic event has a probability of 1/n.
• If event A includes m outcomes, then P(A) = m/n.
• Geometric probability model: Probability proportional to length, area, or volume.
• Geometric probability characteristics: infinite possible outcomes and equally likely outcomes.
• Geometric probability calculation: P(A) = (measure of event A's region) / (measure of the entire sample space).

Limits

• L'Hôpital's Rule: Differentiate numerator and denominator separately until limit type is directly solvable.
• L'Hôpital's Rule applicable types: 0/0 or ∞/∞.
• Techniques for 0 * ∞ or ∞/∞ indeterminate form of limits: transform to eliminate factors, L'Hôpital's Rule, variable substitution, or infinitesimal replacement.
• 0^∞ type limits: similar methods as above, transform to 0/0 or ∞/∞ before applying L'Hôpital's Rule. Complex factors usually become the numerator, especially those containing logarithms.
• ∞-∞ type limits: transform to 0/0 or ∞/∞. Use common denominator if difference of fractions; rationalize if radicals are present.
• Two important limits: lim (x→0) sin(x)/x = 1 and lim (x→∞) (1 + 1/x)^x = e.

Series

• Convergence/divergence definition: If the limit of partial sums exists (S), the series converges.
• If the series converges to S, then S is the sum of the series and rn = S - Sn is the remainder.
• Geometric series: converges if |q| < 1, diverges if |q| ≥ 1.
• P-series: converges if p > 1, diverges if p ≤ 1.
• Basic properties of number series: If Σun converges to S and k is constant, Σkun also converges to kS
• If Σun converges to α and Σvn converges to β, then Σ(un ± vn) converges to α ± β
• Adding, removing, or changing a finite number of terms does not affect convergence
• Squeeze Theorem: If un ≤ vn ≤ wn, and Σun and Σwn converge to α, Σvn also converges to α.
• If Σun and Σwn converge; Σvn may or may not converge.
• Necessary condition for convergence: If Σun converges, then lim (n→∞) un = 0.
• Cauchy convergence criterion: Σun converges iff for any ε > 0, there's N such that if n > N implies |un+1 + un+2 + ... + un+p| < ε for all positive integers p.
• Positive term series: Given series Σ un, if un ≥ 0 for all n, then Σun is a positive term series
• A monotonic and bounded sequence of partial sums is a sufficient and necessary condition.
• Comparison test: Given positive term series Σun and Σvn with un ≤ vn.
• If Σvn converges, so does Σun; if Σun diverges, so does Σvn.
• Ratio test: Let ρ = lim (n→∞) un+1/un. If ρ < 1, the series converges; if ρ > 1 or the limit tends to infinity, it diverges; if ρ = 1, the test is inconclusive.
• Root test: converges if lim (n→∞) (un)^(1/n) < 1 and diverges if limit > 1; inconclusive if limit is 1.
• Alternating series test (Leibniz's theorem): For alternating series Σ(-1)^n un where un ≥ un+1 and lim (n→∞) un = 0 then the series converges.
• Also holds true that |s| ≤ u₁ , and the absolute value of the remainder | rn |≤ U**n+1.
• Absolute convergence: If Σ|un| converges, Σun also converges.
• If Σ|un| converges, it is called absolutely convergent.
• If Σ|un| diverges, but Σun converges, it is called conditionally convergent.

Definite Integrals - Properties

• ∫[a to a] f(x) dx = 0.
• ∫[a to b] dx = b - a.
• ∫[a to b] f(x) dx = -∫[b to a] f(x) dx.
• ∫[a to b] kf(x) dx = k∫[a to b] f(x) dx.
• ∫[a to b] f(x) dx = ∫[a to c] f(x) dx + ∫[c to b] f(x) dx.
• ∫[a to b] [f(x) + g(x)] dx = ∫[a to b] f(x) dx + ∫[a to b] g(x) dx.
• If f(x) ≥ 0 on [a, b], then ∫[a to b] f(x) dx ≥ 0.
• If f(x) ≤ g(x) on [a, b], then ∫[a to b] f(x) dx ≤ ∫[a to b] g(x) dx.
• |∫[a to b] f(x) dx| ≤ ∫[a to b] |f(x)| dx.
• If m ≤ f(x) ≤ M for x in [a, b], then m(b - a) ≤ ∫[a to b] f(x) dx ≤ M(b - a).
• Mean value theorem for integrals: There exists ξ in [a, b] such that ∫[a to b] f(x) dx = f(ξ)(b - a). f(x) must be continuous in [a,b]
• If f(x) is odd, then ∫[-a to a] f(x) dx = 0; if f(x) is even, then ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx.

Indefinite Integrals

• F'(x) = f(b(x))b'(x) – f(a(x))a'(x), where F(x) is integral[a(x), b(x)] of f(t)dt.

Determinants

• Determinant equals its transpose.
• Interchanging two rows/columns changes the sign.
• If two rows/columns are identical, the determinant is zero.
• A common factor in a row/column can be factored out.
• If a row/column is all zeros, the determinant is zero.
• If two rows/columns are proportional, determent value is zero

Matrices

• Matrix: m x n rectangular array of numbers.
• Equal matrices: same dimensions and corresponding elements are equal.
• n-order square matrix if dimensions are nxn
• Zero matrix: all elements are zero.
• Symmetric matrix: aij = aji.
• Anti-symmetric matrix: aij = -aji (diagonal elements are zero).
• Triangular matrix: Upper triangular has zeros below diagonal; lower triangular has zeros above.
• Diagonal matrix: non-diagonal elements are zero.

Linear spaces

• Defined by addition and scalar multiplication operations satisfying specific rules.
• Includes vector addition, scalar multiplication, zero element, negative element, and associativity/commutativity.
• Linear Combination: Linear combinations of vectors α₁, α₂, ..., αr (r ≥ 1) are created when there are constants k₁, …,k*, such that α = k₁α₁ + k₂α₂ + … + k α
• Called the representation of the individual vector, a, by the combination of vector pairs
• Linear Dependence: Linear dependence occurs within linear space v of within vectors α₁, α₂, ..., αr (r 1) when there are constants k₁, …,k* (not all 0) such that k₁α₁ + k₂α₂ + … + k α=0.
• This holds true when k = k₂=....=0
• Dimension: n-demensional if the maximum number of linear independent vectors is n
• Zero vector is not linearly independent
• Linear Subspace: a non empty subset that holds true for two conditions
• Given any vector a in space, w, the scalar product k* a* is present where K is a linear subspace w
• Given any vector space, a, and b, the vector sum a + b is a linear subspace w
• Maximal set of linearly independent vectors is equivalent to itself
• A row that is already equal with the maximal linearily 开发者_如何学Python