Actuary

P(A∪B)= P(A)+P(B)-P(A∩B) P(A∩B) = P(A)+P(B)-P(A∪B) For mutual exclusion, P(A∪B)= P(A)+P(B) nCk P(A|B) = P(A ∩ B) / P(B) Law of Total Probability Bayes' Theorem cumulative distribution function F(x)=Pr(X≤x) Probability Density Function (PDF) f(x)=dF(x)/dx Probability of Ranges for Random Variables Pr(aX≤b)=F(b)-F(a) E(x) for discrete random variables Σxp(x) E(x) for continuous random variables E(min(x,k)) for continuous random variables integral from negative infinity to k [xf(x)dx]+k(1-F(k)) If f(x)=0 for x0, then E(X)= Integral from 0 to infinity (1-F(x)) dx If f(x)=0 for x0, then E(min(X,k))= integral from 0 to k (1-F(x))dx Mean is the same as average ∫ 0 to ∞ xe^(-ax) dx = 1/a^2 Integration by parts ∫ u dv = uv - ∫ v du Mode is the value for which the probability is maximized For independent random variables X and Y, E(XY)= E(X)E(Y) kth raw moment E[X^k] kth central moment E[(X-μ)^k] Variance in terms of moments E[(X-μ)^2]=E[X^2]-μ^2 standard deviation (σ) the square root of the variance coefficient of variation σ/μ Skewness equation E[(X-μ)^3]/σ^3 Kurtosis E[(X-μ)^4]/σ^4 Variance of Linear Function Var(aX+b)= a^2Var(X) Bernoulli Shortcut If Pr(X=a)=1-p and Pr(X=b)=p, then Var(X)=((b-a)^2)p(1-p) Discrete uniform is __________, whereas continuous uniform is _________ integers, all real numbers continuous uniform probability distribution f(x)=1/(b-a) discrete uniform probability distribution f(x)=1/(b-a+1) E(X)= (a+b)/2 Var(X)= (((b-a+1)^2)-1)/12 Cov(X,Y) E(XY) - E(X)E(Y) Independence implies covariance= _____, but _______ 0, not conversely var(aX+bY)= a^2Var(x)+b^2Var(Y)+2abCov(X,Y) Cov(X,Y) = (demonstrates a property) Cov(Y,X) Cov(aX,bY) abCov(X,Y) Cov(X, aY+bZ)= aCov(X,Y) +bCov(X,Z) Corr(X,Y)=ρ= Cov(X,Y)/SD(X)SD(Y) Correlation of -1 or 1 means linear relationship between X and Y