STAT 3704 – Engineering Statistics Midterm 1 Comprehensive Formula & Concept Sheet
====================== DESCRIPTIVE STATISTICS ======================
x̄ = Σxᵢ/n | s² = Σ(xᵢ−x̄ )²/(n−1) | s = √s² | Range = xₘ ₐₓ − xₘ ᵢₙ | IQR = Q₃−Q₁ | Fences = Q₁−1.5(IQR), Q₃+1.5(IQR)
Right skew→mean>median | Left skew→mean<median | Coeff. of Var. CV = s/x̄ ×100% | Empirical Rule: 68%-95%-99.7%
Chebyshev: P(|X−μ|<kσ)≥1−1/k²
====================== VARIABLE TYPES & GRAPHS ======================
Categorical(labels)→bar/pie | Discrete(counts)→stem/hist | Continuous(measured)→hist/box | Normal plot≈straight→normality
====================== PROBABILITY & COMBINATORICS ======================
0≤P(A)≤1 | P(S)=1 | Complement: P(A′)=1−P(A) | Addition: P(A∪B)=P(A)+P(B)−P(A∩B)
Conditional: P(A|B)=P(A∩B)/P(B) | Multiplication: P(A∩B)=P(A|B)P(B) | Independent: P(A∩B)=P(A)P(B)
Mutually exclusive: P(A∩B)=0→P(A∪B)=P(A)+P(B) | Law of Total Prob: P(A)=ΣP(A|Bᵢ)P(Bᵢ)
Bayes’: P(Bᵢ|A)=[P(A|Bᵢ)P(Bᵢ)]/ΣP(A|Bⱼ )P(Bⱼ )
“At least one”=1−P(none) | “At least r”=1−P(X≤r−1) | “At most r”=Σ₀ʳP(X=k) | “Exactly r”=P(X=r)
Symmetry: P(Z>z)=P(Z<−z) | Combination nCr=n!/(r!(n−r)!) | Permutation nPr=n!/(n−r)!
====================== EXPECTATION & VARIANCE RULES ======================
E[aX+b]=aE[X]+b | Var[aX+b]=a²Var[X] | E[X+Y]=E[X]+E[Y] | Var(X+Y)=Var(X)+Var(Y)(indep.)
Cov(X,Y)=E[(X−μₓ)(Y−μᵧ)] | Corr(X,Y)=Cov(X,Y)/(σₓσᵧ)
====================== DISCRETE RANDOM VARIABLES ======================
PMF: Σf(x)=1,f(x)≥0 | E[X]=Σx f(x) | E[X²]=Σx² f(x) | Var(X)=E[X²]−(E[X])² | σ=√Var(X)
From CDF: f(xᵢ)=F(xᵢ)−F(xᵢ₋₁)
====================== BINOMIAL ======================
X~Binomial(n,p): P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ | E=np | Var=np(1−p)
Approx→Poisson if n large,p small (μ=np) | Approx→Normal if np(1−p)>5 → Z=(X−np)/√(np(1−p))
====================== POISSON ======================
X~Poisson(μ): P(X=k)=e^(−μ)μᵏ/k! | E=μ | Var=μ | Additive: Poisson(μ₁)+Poisson(μ₂)=Poisson(μ₁+μ₂)
====================== EXPONENTIAL ======================
X~Exp(λ): f(x)=λe^(−λx),F(x)=1−e^(−λx) | E=1/λ | Var=1/λ²
P(X<a)=1−e^(−λa),P(X>a)=e^(−λa) | Memoryless: P(X>t+s|X>t)=P(X>s)
====================== UNIFORM ======================
X~U(a,b): f(x)=1/(b−a) | E=(a+b)/2 | Var=(b−a)²/12
====================== CONTINUOUS RV ======================
P(a<X<b)=∫ₐᵇf(x)dx | F(x)=∫₋∞ˣf(t)dt | E[X]=∫x f(x)dx | Var(X)=∫(x−E[X])² f(x)dx
====================== NORMAL ======================
X~N(μ,σ): f(x)=1/(σ√2π)e^(−½((x−μ)/σ)²) | Z=(X−μ)/σ | P(a<X<b)=Φ(zb)−Φ(za)
Critical Z: 90%=1.645 | 95%=1.96 | 99%=2.576 | Sym: P(Z<−z)=1−P(Z<z)
Empirical: 68%-95%-99.7% rule
====================== CENTRAL LIMIT THEOREM ======================
For n≥30 or pop Normal: X̄ ~N(μ,σ/√n) | Z=(X̄ −μ)/(σ/√n)
Sample proportion: p̂ ~N(p,√(p(1−p)/n))
====================== t-DISTRIBUTION ======================
t=(X̄ −μ)/(s/√n),df=n−1 | Mean=0,Var=v/(v−2) | Use if σ unknown | n→∞→t→Z
====================== CONFIDENCE INTERVALS ======================
σ known: X̄ ±zₐ/₂(σ/√n) | σ unknown: X̄ ±tₐ/₂,n−1(s/√n)
Proportion: p̂ ±zₐ/₂√(p̂ (1−p̂ )/n) | α=1−CL | width↓as n↑
====================== HYPOTHESIS TESTING ======================
1)H₀/Hₐ 2)α 3)calc stat 4)compare crit/p 5)conclude | t=(X̄ −μ₀)/(s/√n)
Reject H₀ if |t|>tₐ/₂,df | Left tail→< | Right tail→>
====================== SAMPLING & CLT NOTES ======================
Population=all; Sample=subset; Random→equal chance | Statistic(x̄ ,s),Parameter(μ,σ)
As n↑→sample mean≈Normal regardless of distribution
====================== KEY DISTRIBUTIONS ======================
Binomial(n,p):E=np,Var=np(1−p) | Poisson(μ):E=μ,Var=μ | Exp(λ):E=1/λ,Var=1/λ²
Uniform(a,b):E=(a+b)/2,Var=(b−a)²/12 | Normal(μ,σ):E=μ,Var=σ² | t(v):Mean=0,Var=v/(v−2)
====================== SHORTCUTS & CRITICAL VALUES ======================
Complement→1−P | OR→Add−Overlap | AND→Multiply | Indep→P(A)P(B) | MutEx→Add | ≥1→1−P(0)
z=(x−μ)/σ | σ known→Z | σ unknown→t | p<α→Reject H₀ | Round 4dp | Check 0≤P≤1
Z₀.₀₅=1.645 | Z₀.₀₂₅=1.96 | Z₀.₀₀₅=2.576 | t₀.₀₂₅,15=2.131
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