Taking too long? Close loading screen.

SOA

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 …

Read more

SOA ASA Exam: Probability (P)

Basic Calculus Integrations \(\dfrac{d}{dx}a^x=a^x\ln(a)\) \(\int{{{}\over{}}a^xdx}=\dfrac{a^x}{\ln(a)}\text{ for }a>0\)   Logarithmic Differentiation \(\dfrac{df(x)}{dx}=f(x)(\dfrac{d\ln f(x)}{dx})\), since \(\dfrac{d\ln f(x)}{dx}=\dfrac{df(x)/dx}{f(x)}\)   Partial Fraction Decomposition \(\int{{{x}\over{1+x}}dx}=\int{(1-{{1}\over{1+x}})dx}\)   Integration by Parts \(\int{udv}=uv-\int{vdu}\) Special Cases: \(\int_{0}^{\infty }{xe^{-ax}dx}=\dfrac{1}{a^2}\), for \(a>0\) \(\int_{0}^{\infty }{x^2e^{-ax}dx}=\dfrac{2}{a^3}\), for \(a>0\)   Sets Set Properties Associative Property \((A\cup B)\cup C=A\cup (B\cup C)\) and \((A\cap B)\cap C=A\cap (B\cap C)\) Distributive Property \(A\cup (B\cap C)=(A\cup B)\cap (A\cup C))\) and \(A\cap (B\cup C)=(A\cap B)\cup (A\cap C)\) Distributive Property for Complement \((A\cup B)’=A’\cap B’\) and \((A\cap B)’=A’\cup B’\)   Basic Probability Relationships \(Pr[A\cup B]=Pr[A]+Pr[B]-Pr[A\cup B]\) \(Pr[A\cup B\cup C]=Pr[A]+Pr[B]+Pr[C]-Pr[A\cap B]-Pr[B\cap C]-Pr[A\cap C]+Pr[A\cap B\cap C]\) If A and B are independent, then: \(Pr(A\cap B)=Pr(A)Pr(B)\) \(Pr(A\cap B’)=Pr(A)Pr(B’)=Pr(A)(1-Pr(B))\) \(Pr(A’\cap B)=Pr(A’)Pr(B)=(1-Pr(A))Pr(B)\)   Combinatorics Number of Permutations \(n!\)   Number of Distinct Permutations \(_{n}P_k=\dfrac{n!}{(n-k)!}=\dfrac{n!}{{{k}_{1}}!{{k}_{2}}!\cdots {{k}_{j}}!}\)   Number of Combinations \(_{n}C_k=\left( \begin{matrix}n\\k \\\end{matrix} \right)=\dfrac{n!}{k!(n-k)!}=\dfrac{n(n-1)\cdots (n-k+1)}{k!}\)   Conditional Probabilities \(P[A|B]=\dfrac{P[A\cap B]}{P[B]}\)   Bayes’ Theorem Law of Total Probabilities \(P[A]=\sum\limits_{i=1}^{n}{P[B_i]P[A|B_i]}\)   Bayes’ Theorem \(P[A_j|B]=\dfrac{P[A_j]P[B|A_j]}{\sum\limits_{i=1}^{n}{P[A_i]P[B|A_i]}}\)   Random Variables Probability Mass Function (PMF) \(p(x)=Pr(X=x)\)   Probability Density Function (PDF) \(f(x)=\dfrac{dF(x)}{dx}\) or logarithmic differentiation: Given \(F(x)=a\) Taking log: \(\ln F(x) = ln(a)\) Differentiate: \(\dfrac{d\ln F(x)}{dx}=\dfrac{d\ln a}{dx}=\dfrac{d F(x) / dx)}{F(x)}\) Replace the \(F(x)\) in the differentiated formula: \(\dfrac{d \boxed{F(x)}}{dx}=\boxed{F(x)}\times \dfrac{d\ln a}{dx}\) i.e. \(\dfrac{d \boxed{F(x)}}{dx}=\boxed{F(x)}(\dfrac{d\ln \boxed{F(x)}}{dx})\) Similarly, \(\dfrac{d \boxed{f(x)}}{dx}=\boxed{f(x)}(\dfrac{d\ln \boxed{f(x)}}{dx})\) The pdf \(f(x)\) must …

Read more

SOA ASA Exam: Long-Term Actuarial Mathematics (MLC/LTAM)

[mathjax] Review Basic Concepts – Integrals: \(\int_{0}^{\infty }{{{t}^{n}}{{e}^{-ct}}dt}=\dfrac{n!}{{{c}^{n+1}}}\) \(\int_{0}^{u}{{{t}^{n}}{{e}^{-ct}}dt}=\dfrac{1-(1+cu){{e}^{-cu}}}{{{c}^{2}}}\) \({{(\bar{I}\bar{a})}_{u}}=\dfrac{{{{\bar{a}}}_{\overline{u}}}-u{{v}^{u}}}{\delta }\) – Geometric Series: \(\sum\nolimits_{k=0}^{n-1}{a{{r}^{k}}}=a\dfrac{1-{{r}^{n}}}{1-r}\) \({{i}^{(m)}}=m({{(1+i)}^{{}^{1}/{}_{m}}}-1)\) \({{d}^{(m)}}=m(1-{{(1+i)}^{-{}^{1}/{}_{m}}})\) – Survival Function: \({{S}_{x}}(0)=1\) \(\underset{t\to \infty }{\mathop{\lim }}\,{{S}_{x}}(t)=0\) \({{S}_{x}}(t)\) must be a non-increasing function of t Review from MFE – Rate of Discount: \(d=\dfrac{i}{1+i}\) – Discounting Rate: \(v=\dfrac{1}{1+i}=1-d\) – Continuously Compounded Interest Rate: \(\delta =\ln (1+i)\)  – Simple Interest Rate i: \({i}_{t}=1+it\) , \({{v}_{t}}=\dfrac{1}{1+it}\) – PV of an n-Year Immediate Certain Annuity: \({{a}_{\left. {\overline {n}} \right| }}=\dfrac{1-{{v}^{n}}}{i}\) – PV of an n-Year Immediate Certain Annuity-Due: \({{\ddot{a}}_{\left. \overline{n} \right|}}=\dfrac{1-{{v}^{n}}}{d}\)  – PV of an n-Year Continuous Certain Annuity-Due: \({{\bar{a}}_{\left. \overline{n} \right|}}=\dfrac{1-{{v}^{n}}}{\delta}\) – Cumulative Value of an Annuity, the Value at the End of n Years\({{S}_{\left. {\overline {n}}\! \right| }}={{(1+i)}^{n}}{{a}_{\left. {\overline {n}}\! \right| }}\) Survival Function \(\Pr ({{T}_{x}}>t+u)=\Pr ({{T}_{x}}>t)\Pr ({{T}_{x+t}}>u)\) \({{S}_{x}}(t+u)={{S}_{x}}(t){{S}_{x+t}}(u)\) Actuarial Notation \({}_{t+u}{{p}_{x}}={}_{t}{{p}_{x}}{}_{u}{{p}_{x+t}}\) \({}_{\text{t}|u}{{q}_{x}}={}_{t}{{\text{p}}_{x}}-{}_{t+u}{{p}_{x}}={}_{t+u}{{q}_{x}}-{}_{t}{{q}_{x}}\) Life Table \({{d}_{x}}={{l}_{x}}-{{l}_{x+1}}\) \({}_{t}{{p}_{x}}=\dfrac{{{l}_{x+t}}}{{{l}_{x}}}\), \({{q}_{x}}=\dfrac{{{d}_{x}}}{{{l}_{x}}}\) \({}_{\text{t}|u}{{q}_{x}}=\dfrac{{{l}_{x+t}}-{{l}_{x+t+u}}}{{{l}_{x}}}\) Survival Function: Moments Complete Lifetime Complete Life Expectancy \({{\overset{\scriptscriptstyle\smile}{e}}_{x}}=\int_{0}^{\infty }{_{t}{{p}_{x}}dt}\), \(E[{{T}_{x}}^{2}]=2\int_{0}^{\infty }{{{t}_{t}}{{p}_{x}}dt}\) n-Yr. Temporary Life Expectancy \({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}=E[\min ({{T}_{x}},n)]=\int_{0}^{n}{_{t}{{p}_{x}}dt}\) \(E[{{(\min ({{T}_{x}},n))}^{2}}]=2\int_{0}^{n}{t{}_{t}{{p}_{x}}dt}\) Special Mortality Laws Constant Force of Mortality \({{\overset{\scriptscriptstyle\smile}{e}}_{x}}={{\mu }^{-1}}\), \(Var({{T}_{x}})={{\mu }^{-2}}\) \({{\overset{\scriptscriptstyle\smile}{e}}_{x:\left. {\overline {n}}\! \right| }}=\dfrac{1-{{e}^{-\mu n}}}{\mu }\) Uniform & Beta (Derive) For any case in which mortality is uniformly distributed throughout the temporary period, \({{\overset{\smile }{\mathop{e}}\,}_{x:\left. {\overline {n}}\! \right| }}={{n}_{n}}{{p}_{x}}+{{(n/2)}_{n}}{{q}_{x}}\) \({{\overset{\smile }{\mathop{e}}\,}_{x:\left. {\overline {n}}\! \right| }}={{n}_{n}}{{p}_{x}}+{{(n/2)}_{n}}{{q}_{x}}=n(\dfrac{w-x-n}{w-x})+\dfrac{n}{2}(\dfrac{n}{w-x})\) Curtate Lifetime …

Read more