H

Product Rule ;; If task 1 has n₁ ways and task 2 has n₂ ways, total = n₁ × n₂ ||| Sum Rule ;; If mutually exclusive tasks have n₁ or n₂ ways, total = n₁ + n₂ ||| Inclusion-Exclusion (Two Sets) ;; |A ∪ B| = |A| + |B| − |A ∩ B| ||| Pigeonhole Principle ;; k+1 objects into k boxes → at least one box has ≥2 objects ||| Generalized Pigeonhole Principle ;; N objects into k boxes → at least one box has N/k⌉ objects ||| r-Permutations (no repetition) ;; P(n,r) = n! / (n−r)! ||| r-Permutations (with repetition) ;; nʳ ||| r-Combinations (no repetition) ;; C(n,r) = n! / [r!(n−r)!] ||| r-Combinations (with repetition) ;; C(n+r−1, r) = (n+r−1)! / [r!(n−1)!] ||| Binomial Theorem ;; (x+y) = Σₖ₌₀ⁿ C(n,k) xⁿ⁻ᵏ y ||| Pascal's Identity ;; C(n+1,k) = C(n,k−1) + C(n,k) ||| Multinomial Coefficient ;; Distributing n distinct objects into k distinct boxes with nᵢ in box i: n! / (n₁! n₂! … nₖ!) ||| Distinguishable balls → Distinguishable boxes (empty allowed) ;; nʳ ||| Distinguishable balls → Distinguishable boxes (no empty) ;; n! × S(r,n) ||| Indistinguishable balls → Distinguishable boxes (empty allowed) ;; C(r+n−1, r) ||| Indistinguishable balls → Distinguishable boxes (no empty) ;; C(r−1, n−1) ||| Stirling Numbers S(r,i) Formula ;; S(r,i) = (1/i!) Σₖ₌₀ⁱ (−1)ᵏ C(i,k) (i−k)ʳ ||| Sample Space Definition ;; Set of all possible outcomes of an experiment, denoted S ||| Event Definition ;; Any subset E of the sample space S ||| Classical Probability Formula ;; P(E) = n(E) / n(S) where outcomes are equally likely ||| Sure Event ;; P(E) = 1 ||| Impossible Event ;; P(E) = 0 ||| Simple Event ;; Consists of only a single outcome ||| Compound Event ;; Consists of two or more outcomes ||| Mutually Exclusive Events ;; P(A ∩ B) = 0; P(A ∪ B) = P(A) + P(B) ||| Exhaustive Events ;; P(E₁) + P(E₂) + … + P(E) = 1 ||| Independent Events ;; P(A ∩ B) = P(A) × P(B) ||| Dependent Events ;; P(A ∩ B) = P(A) × P(B|A) ||| Complementary Events ;; P(E') = 1 − P(E) ||| Axiom 1 of Probability ;; 0 ≤ P(E) ≤ 1 ||| Axiom 2 of Probability ;; P(S) = 1 ||| Axiom 3 of Probability ;; For mutually exclusive events Eᵢ: P(∪Eᵢ) = ΣP(Eᵢ) ||| Conditional Probability ;; P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0 ||| Multiplication Rule ;; P(A ∩ B) = P(A|B) × P(B) = P(B|A) × P(A) ||| Law of Total Probability ;; P(A) = Σ P(B) × P(A|Bᵢ) where Bᵢ partition S ||| Bayes' Theorem ;; P(B|A) = [P(B) × P(A|B)] / P(A) ||| Generalized Bayes' Theorem ;; P(Bᵢ|A) = [P(B) × P(A|Bᵢ)] / Σⱼ [P(B) × P(A|Bⱼ)] ||| Mean (Sample) ;; x̄ = Σxᵢ / n ||| Median Definition ;; Middle value when sorted; if even n, average two middle terms ||| Mode Definition ;; Value(s) that occur most frequently ||| Population Variance ;; σ² = Σ(xᵢ − μ)² / N ||| Sample Variance ;; s² = Σ(xᵢ − x̄)² / (n−1) ||| Standard Deviation ;; σ = √σ² or s = √s² ||| Random Variable Definition ;; Function assigning a real number to each outcome of a random experiment ||| Discrete RV ;; Takes countable values (finite or countably infinite) ||| Continuous RV ;; Takes uncountably infinite values within intervals ||| PMF Conditions ;; Σpᵢ = 1 and 0 ≤ pᵢ ≤ 1 ||| PDF Conditions ;; f(x) ≥ 0 and ∫₋∞^∞ f(x)dx = 1 ||| CDF Definition ;; F(x) = P(X ≤ x) ||| PDF-CDF Relationship (Continuous) ;; f(x) = d/dx F(x) ||| Expected Value (Discrete) ;; E[X] = Σ xᵢ × P(xᵢ) ||| Expected Value (Continuous) ;; E[X] = ∫₋∞^ x × f(x) dx ||| Variance via Expectation ;; Var(X) = E[X²] − (E[X])² ||| Linearity of Expectation ;; E[aX + b] = aE[X] + b ||| Expectation of Sum ;; E[X + Y] = E[X] + E[Y] ||| Expectation of Constant ;; E[c] = c ||| Expectation of Independent Variables ;; E[XY] = E[X]E[Y] ||| Variance Scaling ;; Var(aX + b) = a²Var(X) ||| Variance of Independent Variables ;; Var(X + Y) = Var(X) + Var(Y) ||| Conditional PMF (Discrete) ;; p_{X|Y}(x|y) = p_{X,Y}(x,y) / p_Y(y) ||| Conditional PDF (Continuous) ;; f_{X|Y}(x|y) = f_{X,Y}(x,y) / f_Y(y) ||| Conditional Expectation (Discrete) ;; E[X|Y=y] = Σ x × p_{X|Y}(x|y) ||| Conditional Expectation (Continuous) ;; E[X|Y=y] = ∫ x × f_{X|Y}(x|y) dx ||| Conditional Variance ;; Var(X|Y=y) = E[X²|Y=y] − (E[X|Y=y])² ||| Tower Property ;; E[E[X|Y]] = E[X] ||| Law of Total Expectation (Discrete) ;; E[X] = Σ_y E[X|Y=y] × P(Y=y) ||| Law of Total Expectation (Continuous) ;; E[X] = E[X|Y=y] × f_Y(y) dy ||| Law of Total Variance (Eve's Law) ;; Var(X) = E[Var(X|Y)] + Var(E[X|Y]) ||| Property: Pulling out knowns ;; E[Xg(Y)|Y] = g(Y)E[X|Y] ||| Property: Independence ;; E[X|Y] = E[X] if X and Y independent ||| Discrete Uniform Profile ;; PMF: 1/(b−a+1); Mean: (a+b)/2; Var: [(b−a+1)²−1]/12 ||| Bernoulli Profile ;; PMF: pˣ(1−p)¹⁻ˣ; Mean: p; Var: p(1−p) ||| Geometric Profile ;; PMF: (1−p)ᵏ⁻¹p; Mean: 1/p; Var: (1−p)/p² ||| Binomial Profile ;; PMF: C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; Mean: np; Var: np(1−p) ||| Poisson Profile ;; PMF: λᵏe⁻λ/k!; Mean: λ; Var: λ ||| Continuous Uniform Profile ;; PDF: 1/(b−a); Mean: (a+b)/2; Var: (b−a)²/12 ||| Exponential Profile ;; PDF: λe⁻λˣ (x≥0); Mean: 1/λ; Var: 1/λ² ||| Normal Distribution Profile ;; PDF: (1/√(2πσ²)) e^(-(x-μ)²/(2σ²)); Mean: μ; Var: σ² ||| Standard Normal Profile ;; Z ~ N(0,1); Mean: 0; Var: 1 ||| MGF Definition ;; M_X(t) = E[e^(tX)] ||| Finding Moments from MGF ;; μₙ' = dⁿM_X(t)/dt evaluated at t=0 ||| MGF Uniqueness ;; MGF uniquely determines the distribution if it exists near t=0 ||| Joint PMF Condition ;; Σ_x Σ_y p_{X,Y}(x,y) = 1 ||| Joint PDF Condition ;; ∫ f_{X,Y}(x,y) dx dy = 1 ||| Marginal PMF ;; p_X(x) = Σ_y p_{X,Y}(x,y) ||| Marginal PDF ;; f_X(x) = ∫ f_{X,Y}(x,y) dy ||| Joint Independence Condition ;; f_{X,Y}(x,y) = f_X(x) × f_Y(y) ||| LOTUS (Discrete) ;; E[g(X,Y)] = ΣΣ g(x,y) p_{X,Y}(x,y) ||| LOTUS (Continuous) ;; E[g(X,Y)] = ∫∫ g(x,y) f_{X,Y}(x,y) dx dy ||| Covariance Alternate Formula ;; Cov(X,Y) = E[XY] − E[X]E[Y] ||| Covariance-Variance Relationship ;; Var(X) = Cov(X,X) ||| Pulling Constants from Covariance ;; Cov(cX, Y) = c × Cov(X,Y) ||| Distributive Property of Covariance ;; Cov(X+Y, Z) = Cov(X,Z) + Cov(Y,Z) ||| Symmetry of Covariance ;; Cov(X,Y) = Cov(Y,X) ||| Constants Cannot Covary ;; Cov(X,c) = 0 ||| Correlation Coefficient ;; ρ = Cov(X,Y) / (σ_X × σ_Y) ||| Correlation Range ;; −1 ≤ ρ ≤ 1 ||| Scale-Invariance of Correlation ;; Corr(aX+b, Y) = sign(a) × Corr(X,Y) ||| Zero Correlation ≠ Independence Trap ;; ρ=0 means no LINEAR relationship only (unless jointly Normal) ||| Markov's Inequality ;; P(X ≥ a) ≤ E[X]/a for non-negative X ||| Chebyshev's Inequality ;; P(|X − μ| ≥ kσ) ≤ 1/k² ||| CLT for Sample Mean ;; X ~ N(μ, σ²/n) as n→∞ ||| CLT for Sum ;; S_n ~ N(nμ, nσ²) as n→∞ ||| Point Estimate ;; Single value guess (e.g., x̄ estimates μ) ||| Interval Estimate ;; Range: Point Estimate ± Margin of Error ||| Confidence Interval (known σ) ;; x̄ ± z_(α/2) × (σ/√n) ||| CI Property: Confidence Level ;; Higher confidence → wider interval ||| CI Property: Sample Size ;; Larger n → narrower interval ||| t-Distribution Use Case ;; σ unknown AND small sample (n<30) ||| t-Distribution Shape ;; Symmetric, bell-shaped, heavier tails than Normal ||| t-Distribution Limit ;; Approaches N(0,1) as n→∞ ||| One-Sample T-Test Statistic ;; t = (x̄ − μ) / (s/√n); df = n−1 ||| Two-Sample T-Test Statistic ;; t = (x̄₁ − x̄₂) / [s_p × √(1/n₁ + 1/n₂)]; df = n₁+n₂−2 ||| Pooled Variance ;; s_p² = [s₁²(n₁−1) + s₂²(n₂−1)] / (n₁+n₂−2) ||| Paired T-Test Statistic ;; t = d̄ / (s_d/√n); df = n−1 ||| Chi-Square Statistic ;; χ² = Σ(O − Eᵢ)² / E ||| Chi-Square GOF Degrees of Freedom ;; df = k − 1 ||| Chi-Square Independence Degrees of Freedom ;; df = (r−1)(c−1) ||| Chi-Square Assumption: Expected Frequency ;; Each cell must have Eᵢ ≥ 5 ||| Z-Test Left-Tailed Rejection ;; Reject H₀ if Z < −z_α ||| Z-Test Right-Tailed Rejection ;; Reject H₀ if Z > z_α ||| Z-Test Two-Tailed Rejection ;; Reject H₀ if |Z| > z_(α/2) ||| Z-Test for Proportion ;; Z = (p − p) / √[p(1−p)/n] ||| Z-Test for Two Means ;; Z = (x̄₁ − x̄₂) / √(σ₁²/n₁ + σ₂²/n₂) ||| Z-Test for Two Proportions ;; Uses pooled proportion p̂ = (n₁p̂₁ + n₂p̂₂)/(n₁+n₂)