Statistics 2141A midterm prep

Population

entire set of objects/outcomes

Sample

subset used for inference

Data-collection methods

Retrospective study, Observational study, Designed experiment

Sampling types

Simple random, Stratified, Convenience

Population mean (μ)

μ = (Σ xᵢ) / N → average of all population values

Sample mean (x̄)

x̄ = (Σ xᵢ) / n → average of sample values

Sample variance (s²)

s² = Σ(xᵢ − x̄)² / (n − 1)

Sample standard deviation

s = √s² → spread of sample data

Inter-quartile range

IQR = Q₃ − Q₁ → middle 50 % of data

Outlier rule

Value beyond 1.5 × IQR from Q₁ or Q₃ is an outlier

Addition rule for probabilities

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Conditional probability

P(A | B) = P(A ∩ B) / P(B)

Independence criterion

P(A ∩ B) = P(A) × P(B)

Bayes' Theorem

P(A | B) = [P(B | A) × P(A)] / P(B)

Law of Total Probability

P(B) = Σ P(B | Aᵢ) P(Aᵢ) for partition {Aᵢ}

Random variable

Numerical variable whose value depends on outcome of a random experiment

Multiplication rule

Total outcomes = n₁ × n₂ × ... × nₖ

Permutation formula

P(n, k) = n! / (n − k)!

Combination formula

C(n, k) = n! / [(n − k)! k!]

Properties of a PDF

f(x) ≥ 0 and ∫ f(x) dx = 1

Probability for interval (a < X < b)

P(a < X < b) = ∫ₐᵇ f(x) dx

Cumulative distribution function

F(x) = ∫₋∞ˣ f(u) du

Variance of a continuous RV

Var(X) = ∫ (x − μ)² f(x) dx = E[X²] − μ².

PDF and mean of Uniform(a,b)

f(x) = 1 / (b − a); E[X] = (a + b)/2.

Variance of Uniform(a,b)

Var(X) = (b − a)² / 12.

PDF of Normal(μ, σ²)

f(x) = (1 / (√(2π)σ)) e^{−(x−μ)² / (2σ²)}.

Standardization formula

Z = (X − μ) / σ.

Empirical rule

≈ 68 % within 1 σ, 95 % within 2 σ, 99.7 % within 3 σ.

Lognormal mean and variance

E[X] = e^{θ + ω² / 2}, Var = e^{2θ + ω²}(e^{ω²} − 1).

Gamma distribution pdf and moments

f(x) = (λʳ xʳ⁻¹ e^{−λx}) / Γ(r); E[X] = r/λ; Var = r/λ².

Chi-square distribution

Special Gamma (λ = ½); E[X] = 2r; Var = 4r.

Weibull mean and variance

E[X] = δ Γ(1 + 1/β); Var = δ²[Γ(1 + 2/β) − (Γ(1 + 1/β))²].

Beta distribution mean and variance

E[X] = α / (α + β); Var = αβ / [(α + β)²(α + β + 1)].

Binomial pmf and moments

f(x) = C(n, x)pˣ(1 − p)ⁿ⁻ˣ; E[X] = np; Var = np(1 − p).

Geometric distribution

f(x) = (1 − p)^{x−1} p; E[X] = 1/p; Var = (1 − p)/p².

Negative binomial distribution

f(x) = C(x−1, r−1)(1 − p)^{x−r} pʳ; E[X] = r/p; Var = r(1 − p)/p².

Hypergeometric distribution

f(x) = [C(K, x) C(N − K, n − x)] / C(N, n); E[X] = np; Var = np(1 − p)(N − n)/(N − 1).

Poisson distribution

f(x) = e^{−λ} λˣ / x!; E[X] = λ; Var = λ.

Exponential distribution

f(x) = λ e^{−λx}; E[X] = 1/λ; Var = 1/λ²; memoryless property.

When does Binomial ≈ Poisson?

n large, p small, λ = n p.

When does Binomial ≈ Normal?

np > 5 and n(1−p) > 5; use continuity correction ± 0.5.

When does Poisson ≈ Normal?

λ > 5; use N(λ, λ) with ± 0.5 correction.

Expectation of aX + b

E[aX + b] = a E[X] + b.

Variance of aX + b

Var(aX + b) = a² Var(X).

Additive variance of independent RVs

Var(X + Y) = Var(X) + Var(Y).

Memoryless property belongs to which distribution?

Exponential: P(X ≥ s + t | X ≥ s) = P(X ≥ t).

Central Limit Theorem summary

Sum of many independent RVs ≈ Normal distribution.