Exam P Formula Sheet: Every Formula You Need to Memorize
A comprehensive formula sheet covering every key formula for SOA Exam P.
Probability Fundamentals
P(A union B) = P(A) + P(B) - P(A intersect B). For three events: P(A union B union C) = P(A) + P(B) + P(C) - P(AB) - P(AC) - P(BC) + P(ABC). P(A|B) = P(A intersect B)/P(B). Bayes' Theorem: 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). Complement: P(A^c) = 1 - P(A). De Morgan's laws: (A union B)^c = A^c intersect B^c and (A intersect B)^c = A^c union B^c.
Permutations: P(n,k) = n!/(n-k)!. Combinations: C(n,k) = n!/(k!(n-k)!). Multinomial coefficient: n!/(n1!*n2!*...*nk!). Stars and bars (distributing n identical objects into k bins): C(n+k-1, k-1). Inclusion-exclusion for counting: |A union B| = |A| + |B| - |A intersect B|.
Expectation and Variance
E[X] = sum x*p(x) for discrete, or integral x*f(x) dx for continuous. LOTUS: E[g(X)] = sum g(x)*p(x) or integral g(x)*f(x) dx. Var(X) = E[X^2] - (E[X])^2. Standard deviation: SD(X) = sqrt(Var(X)).
Linear transformations: E[aX + b] = a*E[X] + b. Var(aX + b) = a^2 * Var(X). E[aX + bY] = a*E[X] + b*E[Y] (always, regardless of dependence).
Covariance: Cov(X,Y) = E[XY] - E[X]*E[Y]. Cov(aX+b, cY+d) = ac*Cov(X,Y). Var(X + Y) = Var(X) + Var(Y) + 2*Cov(X,Y). Var(X - Y) = Var(X) + Var(Y) - 2*Cov(X,Y). Correlation: rho = Cov(X,Y)/(sigma_X * sigma_Y), with -1 <= rho <= 1.
For independent X, Y: E[XY] = E[X]*E[Y], Cov(X,Y) = 0, Var(X + Y) = Var(X) + Var(Y), M_{X+Y}(t) = M_X(t)*M_Y(t).
Discrete Distribution Parameters
Bernoulli(p): E = p, Var = p(1-p), MGF = (1-p) + p*e^t.
Binomial(n, p): E = np, Var = np(1-p), MGF = (1-p+p*e^t)^n.
Poisson(lambda): E = Var = lambda, MGF = exp(lambda*(e^t - 1)). Recursive: P(k+1) = (lambda/(k+1))*P(k).
Geometric(p) (trials until first success): E = 1/p, Var = (1-p)/p^2.
Negative Binomial(r, p) (trials until r-th success): E = r/p, Var = r(1-p)/p^2.
Hypergeometric (N items, K successes, n draws): E = nK/N.
Discrete Uniform on {1,...,n}: E = (n+1)/2, Var = (n^2-1)/12.
Continuous Distribution Parameters
Uniform(a, b): E = (a+b)/2, Var = (b-a)^2/12, f(x) = 1/(b-a) for a < x < b.
Exponential(lambda): E = 1/lambda, Var = 1/lambda^2, S(x) = e^(-lambda*x), MGF = lambda/(lambda-t). Memoryless: P(X > s+t | X > s) = P(X > t).
Normal(mu, sigma^2): MGF = exp(mu*t + sigma^2*t^2/2). Z = (X-mu)/sigma ~ N(0,1). Key z-values: z_0.90 = 1.282, z_0.95 = 1.645, z_0.975 = 1.960, z_0.99 = 2.326.
Gamma(alpha, lambda): E = alpha/lambda, Var = alpha/lambda^2, MGF = (lambda/(lambda-t))^alpha. Special cases: Exponential = Gamma(1, lambda); Chi-squared(k) = Gamma(k/2, 1/2).
Pareto(alpha, theta): E = theta/(alpha-1) for alpha > 1, Var = alpha*theta^2/((alpha-1)^2*(alpha-2)) for alpha > 2, S(x) = (theta/(theta+x))^alpha.
Lognormal(mu, sigma^2): E = exp(mu + sigma^2/2), Var = exp(2*mu + sigma^2)*(exp(sigma^2) - 1), P(X <= x) = Phi((ln(x) - mu)/sigma).
Beta(a, b): E = a/(a+b), Var = ab/((a+b)^2*(a+b+1)), support (0, 1).
Weibull(theta, tau): S(x) = exp(-(x/theta)^tau), h(x) = (tau/theta)*(x/theta)^(tau-1).
Conditional Probability and Bayes
Law of total probability: P(A) = sum P(A|B_i)*P(B_i). Law of total expectation (tower property): E[X] = E[E[X|Y]]. Law of total variance: Var(X) = E[Var(X|Y)] + Var(E[X|Y]). For bivariate normal: E[Y|X=x] = mu_Y + rho*(sigma_Y/sigma_X)*(x - mu_X), Var(Y|X=x) = sigma_Y^2*(1 - rho^2).
Insurance and Actuarial Formulas
Deductible d: payment per loss Y = (X - d)+ = max(X - d, 0). E[(X-d)+] = E[X] - E[min(X, d)]. E[min(X, d)] = integral from 0 to d of S(x) dx. For exponential: E[(X-d)+] = theta*e^(-d/theta). Loss elimination ratio: LER = E[min(X, d)] / E[X].
Policy limit u: payment = min(X, u). E[min(X, u)] = integral from 0 to u of S(x) dx. Combined deductible and limit: payment = min(X, d+u) - min(X, d).
Compound model: S = X_1 + ... + X_N. E[S] = E[N]*E[X]. Var(S) = E[N]*Var(X) + Var(N)*(E[X])^2. Poisson frequency shortcut: Var(S) = lambda*E[X^2].
Mean excess function: e(d) = E[X - d | X > d]. E[(X-d)+] = e(d)*P(X > d). For exponential: e(d) = theta (constant). For Pareto: e(d) = (theta + d)/(alpha - 1) (increasing).
Key Theorems and Inequalities
CLT: for large n, (X_bar - mu)/(sigma/sqrt(n)) is approximately N(0, 1). Equivalently, S_n is approximately Normal(n*mu, n*sigma^2). Continuity correction: for discrete variables, P(S >= k) uses k - 0.5.
Survival function method: E[X] = integral from 0 to infinity of S(x) dx for X >= 0. Hazard rate: h(x) = f(x)/S(x) = -d/dx ln(S(x)). S(x) = exp(-integral from 0 to x of h(t) dt).
Order statistics from n iid variables with CDF F: f_{X_(k)}(x) = n!/((k-1)!(n-k)!) * F(x)^(k-1) * (1-F(x))^(n-k) * f(x). Minimum: F_{X_(1)}(x) = 1 - (1-F(x))^n. Maximum: F_{X_(n)}(x) = (F(x))^n.
Chebyshev: P(|X - mu| >= k*sigma) <= 1/k^2. Markov: P(X >= a) <= E[X]/a for X >= 0. Jensen: E[g(X)] >= g(E[X]) for convex g, E[g(X)] <= g(E[X]) for concave g.
MGF: M(t) = E[e^(tX)]. E[X^n] = M^(n)(0). Uniqueness: same MGF implies same distribution. Sum of independents: M_{X+Y}(t) = M_X(t)*M_Y(t). Linear transformation: M_{aX+b}(t) = e^(bt)*M_X(at).