P Exam Formulas

1-P(A)

P(A’)

P(A OR B)

P(A) + P(B) - P(AB)

Mutually Exclusive Fundamental

P(AB) = 0

P(A | B)

P(AB) / P(B)

P(AB)

P(A) * P(B | A) = P(B) * P(A | B)

Independence

P(AB) = P(A) * P(B) and P(A | B) = P(A)

Law of Total Probability

P(B) = sum(P(B | Ai) * P(Ai))

Bayes’ Theorem

P(Ai | B) = P(AiB) / sum(P(B | Ai) * P(Ai))

Var(X)

E(X²) - E(X)²

Var(cX)

c² * Var(X)

SD(X)

sqrt(Var(X))

Coefficient of Variation

SD(X) / E(X)

CDF

F(x) = P(X <= x)

CDF Discrete

sum[n <= x](P(X = n))

CDF Continuous

int[x, -inf](f(t) dt)

Density

f(t) = d/dx F(x)

Mixed Properties (w/ jump at a)

P(X = a) = F(a) - lim[x^a] F(x)

Discrete Moments

E(X) = sum[x](x * P(X = x))

E(X²) = sum[x](x² * P(X = x))

E(g(X)) = sum[x](g(x) * P(X = x))

Continuous Moments

E(X) = int(x * f(x) dx)

E(X²) = int(x² * f(x) dx)

E(g(X)) = int(g(x) * f(x) dx)

Mixed Moments

Sum discrete and continuous

Survival Moments (X >= 0)

E(X) = int[inf, 0](1-F(x) dx)

E(g(X)) = int[inf,0](g’(x) * (1 - F(x)) dx)

Marginal Distribution of X

P(X = x) = sum[y](P(X = x, Y = y))

P(X = x | Y = y)

P(X = x, Y = y) / P(Y = y)

F(x,y)

P(X <= x, Y <= y)

F(x) (joint distribution)

P(X <= x) = F(x, inf)

Cov(X,Y)

E(XY) - E(X) * E(Y)

Covariance Properties

Cov(X,X) = Var(x)

Cov(aX+ bY, cZ + dW) = acCov(X,Z) + adCov(X,W) + bcCov(Y, Z) + bdCov(Y, W)

Var(aX + bY) = a²Var(X) + 2abCov(X,Y) + b²Var(Y)

Cov(X,Y) (X and Y are indpendent)

0

Var(X + Y) = Var(X) + Var(Y)

Corr(X,Y)

Cov(X,Y) / SD(X) * SD(Y)

Conditional Moments

E(X | Y = y) = sum[x](x * P(X = x | Y = y)

E(X) = E(E(X | Y))

Law of Total Variation

Var(X) = E(Var(X | Y)) + Var(E(X | Y))

Var(S) = E(Y) * Var(X) + E(X)² * Var(Y)

P(min{X1, Xn} > x)

P(Y1 > x) = P(X > x)^n

P(max{X1,Xn} <= x)

P(Yn <= x) = P(X <= x)^n

f(Yi)

i * (n chose i) * F[x](y)^(i-1)