SOA ASA Exam: Short-Term Actuarial Mathematics (STAM/C)
[mathjax] Basic Probability Functions and Moments Probability Density Function \(f(x)=\dfrac{d}{dx}F(x)\) or \(f(x)=-\dfrac{d}{dx}S(x)\) Hazard Rate Function \(\mu (x)=h(x)=\dfrac{f(x)}{S(x)}=-\dfrac{d\ln S(x)}{dx}\) Cumulative Hazard Rate Function \(H(x)=\int_{-\infty }^{x}{h(t)dt}=-\ln S(x)\) \(S(x)=e^{-H(x)}=e^{-\int_{-\infty }^{x}{h(t)dt}}\) Moment of X nth Raw Moment of X: \(\mu’_{n}=E[x^n]\) nth Central Moment of X: \(\mu_n=E[{(x-\mu )}^n]\) Covariance: \(Cov(X,Y)=E[(X-\mu_X)(Y-\mu_Y)]=E[XY]-E[X]E[Y]\) Correlation Coefficient: \(\rho_{XY}=Cov(X,Y)/({\sigma_X}{\sigma_Y})\) Coefficient of Variance: \(\sigma /\mu \) Skewness: \(\gamma_1=\mu_3/\sigma^3\) Kurtosis: \(\gamma_2=\mu_4/\sigma^4\) Moment Generating Function: \(M_X(t)=E[e^{tX}]\) Probability Generating Function: \(P_X(t)=E[t^X]\) Differentials \(\dfrac{d}{dx}{{c}^{ax}}=a\ln (c){{c}^{ax}}\) \(\dfrac{d}{dx}{{\log }_{c}}x=\frac{1}{x\ln (c)}\) Sum of Distributions If \(X_1 \sim Gamma(\alpha_1,\theta)\), …, \(X_n \sim Gamma(\alpha_n,\theta)\), then \(X=\sum{X_i} \sim Gamma(\alpha=\alpha_1+\alpha_2+…+\alpha_n, \theta)\) If \(X_1 \sim Exp(\theta)\), …, \(X_n \sim Exp(\theta)\), then \(X=\sum{X_i} \sim Gamma(\alpha=n, \theta)\) If \(X_1 \sim Poi(\lambda_1)\), …, \(X_n \sim Poi(\lambda_n)\), then \(X=\sum{X_i} \sim Poi(\lambda=\lambda_1+\lambda_2+…+\lambda_n)\) If \(X_1 \sim Bin(m_1,q)\), …, \(X_n \sim Bin(m_n,q)\), then \(X=\sum{X_i} \sim Bin(m=m_1+m_2+…+m_n,q)\) If \(X_1 \sim NB(r_1,\beta)\), …, \(X_n \sim NB(r_n,\beta)\), then \(X=\sum{X_i} \sim NB(r=r_1+r_2+…+r_n,\beta)\) Integration by Parts \(\int{udv}=uv-\int{vdu}\) Conditional probability and expectation Bayes’ Theorem \(\Pr (A|B)=\dfrac{\Pr (B|A)\Pr (A)}{\Pr (B)}\) or \(f_X(x|y)=\dfrac{f_Y(y|x)f_X(x)}{f_Y(y)}\) Law of Total Probability If \(B_i\) is a set of exhaustive (in other words, \(\Pr (\bigcup\nolimits_{i}{B_i})=1\)) and mutually exclusive (in other words \(\Pr (B_i\bigcap{B_j})=0\) for \(i\ne j\)) events, then for any event A, \(\Pr (A)=\sum\limits_{i}{\Pr (A\bigcap{B_i})}=\sum\limits_{i}{\Pr (A|B_i)\Pr (B_i)}\) Correspondingly for continuous distributions, \(\Pr (A)=\int{\Pr (A|x)f(x)dx}\) Conditional …