Exam P Formula Sheet: Every Formula You Need to Memorize

Probability Fundamentals

P(A union B) = P(A) + P(B) - P(A intersect B). P(A|B) = P(A intersect B)/P(B). Bayes: P(B_i|A) = P(A|B_i)*P(B_i) / sum(P(A|B_j)*P(B_j)). Independence: P(A intersect B) = P(A)*P(B). Permutations: P(n,k) = n!/(n-k)!. Combinations: C(n,k) = n!/(k!(n-k)!). Multinomial: n!/(n1!*n2!*...*nk!).

E[X] = sum x*p(x) or integral x*f(x)dx. Var(X) = E[X^2] - (E[X])^2. Var(aX+b) = a^2*Var(X). Cov(X,Y) = E[XY] - E[X]*E[Y]. Var(X+Y) = Var(X) + Var(Y) + 2*Cov(X,Y). Correlation: rho = Cov(X,Y)/(sigma_X*sigma_Y).

Distribution Parameters

Binomial(n,p): E = np, Var = np(1-p), MGF = (1-p+pe^t)^n. Poisson(lambda): E = Var = lambda, MGF = exp(lambda(e^t-1)). Normal(mu,sigma^2): MGF = exp(mu*t + sigma^2*t^2/2). Exponential(lambda): E = 1/lambda, Var = 1/lambda^2, MGF = lambda/(lambda-t). Gamma(alpha,lambda): E = alpha/lambda, Var = alpha/lambda^2, MGF = (lambda/(lambda-t))^alpha. Uniform(a,b): E = (a+b)/2, Var = (b-a)^2/12.

Pareto(alpha,theta): E = theta/(alpha-1), Var = alpha*theta^2/((alpha-1)^2*(alpha-2)). Lognormal(mu,sigma^2): E = exp(mu+sigma^2/2), Var = exp(2*mu+sigma^2)*(exp(sigma^2)-1).

Key Theorems and Insurance Formulas

CLT: (X_bar - mu)/(sigma/sqrt(n)) -> N(0,1). Total expectation: E[X] = E[E[X|Y]]. Total variance: Var(X) = E[Var(X|Y)] + Var(E[X|Y]). Compound model: E[S] = E[N]*E[X], Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. Poisson compound: Var(S) = lambda*E[X^2]. Deductible: E[(X-d)+] = E[X] - E[min(X,d)]. Survival: E[X] = integral S(x)dx. Hazard: h(x) = f(x)/S(x), S(x) = exp(-integral h). Order statistics: f_{X_(k)}(x) = n!/((k-1)!(n-k)!) * F^(k-1) * (1-F)^(n-k) * f(x). Chebyshev: P(|X-mu| >= k*sigma) <= 1/k^2. Markov: P(X >= a) <= E[X]/a. Jensen: E[g(X)] >= g(E[X]) for convex g.