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 …