STAT 411 – Complete Formula and Concept Reference (Ch. 1-11)

A comprehensive set of vocabulary flashcards covering fundamental probability rules, key distributions, estimation, hypothesis testing, regression, and essential calculus rules from STAT 411 Chapters 1–11.

Sample Space

The set of all possible outcomes of an experiment.

Probability Rules

Fundamental properties: P(A) ≥ 0 and P(S) = 1, where S is the sample space.

Addition Rule

For any events A and B, P(A ∪ B) = P(A)+P(B)−P(A ∩ B).

Multiplication Rule (Independent)

If A and B are independent, P(A ∩ B) = P(A)·P(B).

Conditional Probability

The probability of A given B: P(A|B) = P(A ∩ B)/P(B).

Bayes’ Theorem

P(A|B) = [P(B|A)·P(A)]/P(B), reverses conditional probabilities.

Permutations (nPr)

Ordered selections: nPr = n!/(n−r)!.

Combinations (nCr)

Unordered selections: nCr = n!/[r!(n−r)!].

Expected Value E(X)

Long-run mean: Σx·P(x) for discrete X.

Variance Var(X)

Measure of spread: E(X²) − [E(X)]².

Binomial Distribution

Discrete model for number of successes: P(X=k)=nCk p^k (1−p)^(n-k)

Binomial Mean

E(X)=np for Binomial(n,p).

Binomial Variance

Var(X)=np(1−p) for Binomial(n,p).

Poisson Distribution

Counts rare events: P(X=k)=λ^k e^{−λ}/k!.

Poisson Mean & Variance

Both equal λ.

Exponential Distribution

Continuous model for waiting time with rate λ.

Exponential PDF

f(x)=λe^{−λx}, x≥0.

Exponential CDF (Tail)

P(X> x)=e^{−λx}.

Exponential Mean

E(X)=1/λ.

Exponential Variance

Var(X)=1/λ².

Joint PDF f(x,y)

Function giving probability density for pair (X,Y).

Marginal Distribution

Distribution of one variable found by integrating the joint PDF over the other variable.

Independence (Joint)

X and Y independent if f(x,y)=fX(x)·fY(y).

Covariance Cov(X,Y)

E[XY] − E[X]E[Y], measures joint variability.

Correlation ρ

Cov(X,Y)/(σX·σY), standardized measure of linear association.

Confidence Interval for Mean (σ known)

x̄ ± z*·(σ/√n).

Confidence Interval for Mean (σ unknown)

x̄ ± t*·(s/√n).

Confidence Interval for Proportion

p̂ ± z*·√[p̂(1−p̂)/n].

z-test for Mean

(x̄−μ)/(σ/√n), tests population mean with known σ.

t-test for Mean

(x̄−μ)/(s/√n), tests mean when σ unknown.

Proportion z-test

(p̂−p₀)/√[p₀(1−p₀)/n], tests single proportion.

P-value

Probability of observing data as extreme as sample under H₀.

Critical Value Rule

Reject H₀ if test statistic falls in rejection region defined by α.

Two-sample t-test (Unpooled)

t=(x̄₁−x̄₂)/√(s₁²/n₁+s₂²/n₂), unequal variances.

Pooled t-test

Uses pooled variance S_p² and assumes equal variances.

Paired t-test

t=d̄/(s_d/√n), analyzes matched pairs differences.

Two-proportion z-test

z=(p̂₁−p̂₂)/√[p̂(1−p̂)(1/n₁+1/n₂)], where p̂ is pooled proportion.

Simple Linear Regression Model

Y=β₀+β₁X+ε, relates a response to a predictor.

Least Squares Slope β̂₁

β̂₁=Sxy/Sxx, minimizes sum of squared errors.

Least Squares Intercept β̂₀

β̂₀=ȳ−β̂₁x̄.

Coefficient of Determination R²

Proportion of variation explained: SSR/SST.

t-test for Slope

t=β̂₁/SE(β̂₁), tests if β₁=0.

Power Rule (Integration)

∫x^n dx = x^{n+1}/(n+1)+C for n≠−1.

Log Integration Rule

∫x^{−1}dx=ln|x|+C.

Linear Function Integration

∫(ax+b)^n dx=(ax+b)^{n+1}/[a(n+1)]+C.

Exponential Integration

∫e^{ax}dx=(1/a)e^{ax}+C.

Central Limit Theorem (CLT)

For large n, x̄≈N(μ,σ/√n) regardless of population distribution.

Standard Error (SE)

Estimated standard deviation of a statistic, e.g., σ/√n or s/√n.