Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Lesson 1.1: learn & practice →Formula reference
A study companion. Follow each link for the assumptions, worked examples, and practice. This is not an official exam exhibit.
Counting
C(n,r)=n!/[r!(n−r)!]; P(n,r)=n!/(n−r)!
Lesson 1.2: learn & practice →Conditional probability
P(A | B)=P(A ∩ B)/P(B), if P(B)>0
Lesson 1.3: learn & practice →Bayes’ theorem
P(Aᵢ | B)=P(B | Aᵢ)P(Aᵢ)/Σⱼ P(B | Aⱼ)P(Aⱼ), for a partition {Aⱼ}
Lesson 1.5: learn & practice →Expectation
E[g(X)]=Σg(x)p(x); for a density, integrate g(x)f(x)
Lesson 2.2: learn & practice →Variance
Var(X)=E[X²]−E[X]²; Var(aX+b)=a²Var(X)
Lesson 2.3: learn & practice →Binomial
P(X=k)=C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; E[X]=np; Var(X)=np(1−p)
Lesson 2.4: learn & practice →Geometric: trials until success
P(X=k)=p(1−p)ᵏ⁻¹; E[X]=1/p; Var(X)=(1−p)/p²
Lesson 2.5: learn & practice →Poisson
P(X=k)=e^(−λ)λᵏ/k!; E[X]=Var(X)=λ
Lesson 2.6: learn & practice →Continuous probability
P(a<X≤b)=F(b)−F(a); f(x)=F′(x) where differentiable
Lesson 3.1: learn & practice →Exponential: rate λ
P(X>x)=e^(−λx), x≥0; E[X]=1/λ; Var(X)=1/λ²
Lesson 3.2: learn & practice →Standardization
Z=(X−μ)/σ for a normal X; Φ(z)=P(Z≤z)
Lesson 3.3: learn & practice →Calculus & notation readiness
∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f
Lesson 0.1: learn & practice →Joint probability tables
F(a,b) = Σ[x≤a, y≤b] p(x,y)
Lesson 4.1: learn & practice →Marginals & conditional distributions
p(x | y) = p(x,y) / p_Y(y)
Lesson 4.2: learn & practice →Conditional moments
Var(X | Y=y) = E[X² | Y=y] − E[X | Y=y]²
Lesson 4.3: learn & practice →Covariance & correlation
Cov(X,Y) = E[XY] − E[X]E[Y]; ρ = Cov(X,Y)/(σ_Xσ_Y)
Lesson 4.4: learn & practice →Sums & linear combinations
Var(Σ aᵢXᵢ) = Σ aᵢ²Var(Xᵢ), for independent Xᵢ
Lesson 5.1: learn & practice →Order statistics
P(max≤x)=F(x)ⁿ; P(min>x)=[1−F(x)]ⁿ (i.i.d. sample)
Lesson 5.2: learn & practice →Central Limit Theorem
Z = (Sₙ − nμ)/(σ√n); SD(X̄)=σ/√n
Lesson 5.3: learn & practice →Losses, deductibles & limits
Y = min(max(X−d,0),L), where L is maximum payment
Lesson 6.1: learn & practice →Coinsurance & inflation
Y = min(c·max((1+i)X−d,0),L)
Lesson 6.2: learn & practice →Payment moments & risk
E[Y]=Σg(x)p(x); Var(Y)=E[Y²]−E[Y]²; CV=SD(Y)/E[Y]
Lesson 6.3: learn & practice →