Exam P -- Probability
Formula Reference
All formulas tested on SOA Exam P, organized by topic. Use alongside practice problems.
The only material provided during Exam P is a standard normal table. Everything else must be memorized.
General Probability1
Inclusion-Exclusion
\(\Pr(A \cup B) = \Pr(A) + \Pr(B) - \Pr(A \cap B)\)
\(\Pr(A \cup B \cup C) = \Pr(A) + \Pr(B) + \Pr(C)\)
\(\quad - \Pr(A \cap B) - \Pr(B \cap C) - \Pr(A \cap C) + \Pr(A \cap B \cap C)\)
\(\Pr(A \cup B \cup C) = \Pr(A) + \Pr(B) + \Pr(C)\)
\(\quad - \Pr(A \cap B) - \Pr(B \cap C) - \Pr(A \cap C) + \Pr(A \cap B \cap C)\)
Conditional Probability
\(\Pr(A|B) = \dfrac{\Pr(A \cap B)}{\Pr(B)}\)
\(\Pr(A \cap B) = \Pr(A|B)\Pr(B) = \Pr(B|A)\Pr(A)\)
\(\Pr(A \cap B) = \Pr(A|B)\Pr(B) = \Pr(B|A)\Pr(A)\)
Independence
\(\Pr(A \cap B) = \Pr(A)\cdot\Pr(B)\)
\(\Pr(A|B) = \Pr(A)\)
\(\Pr(A'|B) = \Pr(A')\)
\(\Pr(A|B) = \Pr(A)\)
\(\Pr(A'|B) = \Pr(A')\)
Bayes' Theorem
\[\Pr(A_k|B) = \frac{\Pr(B|A_k)\cdot\Pr(A_k)}{\displaystyle\sum_{i=1}^{n}\Pr(B|A_i)\cdot\Pr(A_i)}\]
Law of Total Probability
\(\Pr(B) = \displaystyle\sum_{i=1}^{n}\Pr(B|A_i)\cdot\Pr(A_i)\)
De Morgan's Laws
\(\Pr[(A\cup B)'] = \Pr(A'\cap B')\)
\(\Pr[(A\cap B)'] = \Pr(A'\cup B')\)
\(\Pr[(A\cap B)'] = \Pr(A'\cup B')\)
Combinatorics
\({}_nP_k = \dfrac{n!}{(n-k)!}\) (permutations)
\({}_nC_k = \binom{n}{k} = \dfrac{n!}{(n-k)!\,k!}\) (combinations)
Multinomial: \(\dfrac{n!}{k_1!\,k_2!\cdots k_r!}\)
\({}_nC_k = \binom{n}{k} = \dfrac{n!}{(n-k)!\,k!}\) (combinations)
Multinomial: \(\dfrac{n!}{k_1!\,k_2!\cdots k_r!}\)
Univariate Distribution Fundamentals2
CDF, PDF, Survival
\(F_X(x) = \Pr(X \le x)\)
\(f_X(x) = \tfrac{d}{dx}F_X(x)\)
\(S_X(x) = 1 - F_X(x) = \Pr(X > x)\)
\(\Pr(a < X \le b) = F_X(b) - F_X(a)\)
\(f_X(x) = \tfrac{d}{dx}F_X(x)\)
\(S_X(x) = 1 - F_X(x) = \Pr(X > x)\)
\(\Pr(a < X \le b) = F_X(b) - F_X(a)\)
Expected Value
\(\text{E}[X] = \displaystyle\int_{-\infty}^{\infty} x\,f_X(x)\,dx\)
Alternate: \(\text{E}[X] = \displaystyle\int_0^{\infty} S_X(x)\,dx\) for \(X \ge 0\)
\(\text{E}[g(X)] = \displaystyle\int g(x)\,f_X(x)\,dx\)
Alternate: \(\text{E}[X] = \displaystyle\int_0^{\infty} S_X(x)\,dx\) for \(X \ge 0\)
\(\text{E}[g(X)] = \displaystyle\int g(x)\,f_X(x)\,dx\)
Variance
\(\text{Var}[X] = \text{E}[X^2] - (\text{E}[X])^2\)
\(\text{Var}[aX + b] = a^2\,\text{Var}[X]\)
\(\text{SD}[X] = \sqrt{\text{Var}[X]}\)
\(\text{CV}[X] = \text{SD}[X]/\text{E}[X]\)
\(\text{Var}[aX + b] = a^2\,\text{Var}[X]\)
\(\text{SD}[X] = \sqrt{\text{Var}[X]}\)
\(\text{CV}[X] = \text{SD}[X]/\text{E}[X]\)
Conditional Expectation
\(\text{E}[g(X)|j \le X \le k] = \dfrac{\displaystyle\int_j^k g(x)\,f_X(x)\,dx}{\Pr(j \le X \le k)}\)
Percentile & Mode
100p-th percentile: smallest \(\pi_p\) where \(F_X(\pi_p) \ge p\)
Mode: value(s) where \(f_X(x)\) is maximized
Mode: value(s) where \(f_X(x)\) is maximized
Discrete Distributions3
Complete Reference
| Distribution | PMF | Mean | Variance | Key Property |
|---|---|---|---|---|
| Discrete Uniform |
\(\dfrac{1}{b-a+1},\quad x=a,\ldots,b\) | \(\dfrac{a+b}{2}\) | \(\dfrac{(b-a+1)^2-1}{12}\) | -- |
| Binomial Bin(n,p) |
\(\binom{n}{x}p^x(1-p)^{n-x}\) | \(np\) | \(np(1-p)\) | Sum of indep. Bin same p ~ Bin(\(\sum n_i, p\)) |
| Geometric (# trials) |
\((1-p)^{x-1}p,\quad x=1,2,\ldots\) | \(\dfrac{1}{p}\) | \(\dfrac{1-p}{p^2}\) | Memoryless: \((X-c|X>c)\sim X\) |
| Negative Binomial |
\(\binom{x-1}{r-1}p^r(1-p)^{x-r}\) | \(\dfrac{r}{p}\) | \(r\!\left(\dfrac{1-p}{p^2}\right)\) | Neg Bin(1,p) ~ Geometric(p) |
| Poisson(\(\lambda\)) | \(\dfrac{e^{-\lambda}\lambda^x}{x!},\quad x=0,1,\ldots\) | \(\lambda\) | \(\lambda\) | Sum of indep. Poisson ~ Poisson(\(\sum\lambda_i\)) |
| Hypergeometric | \(\dfrac{\binom{m}{x}\binom{N-m}{n-x}}{\binom{N}{n}}\) | \(\dfrac{nm}{N}\) | \(\dfrac{nm}{N}\cdot\dfrac{N-m}{N}\cdot\dfrac{N-n}{N-1}\) | Sampling without replacement |
Continuous Distributions4
Complete Reference
| Distribution | CDF | Mean | Variance | Key Property | |
|---|---|---|---|---|---|
| Uniform(a,b) | \(\dfrac{1}{b-a}\) | \(\dfrac{x-a}{b-a}\) | \(\dfrac{a+b}{2}\) | \(\dfrac{(b-a)^2}{12}\) | \((X|c |
| Exponential(\(\theta\)) | \(\dfrac{1}{\theta}e^{-x/\theta}\) | \(1-e^{-x/\theta}\) | \(\theta\) | \(\theta^2\) | Memoryless: \((X-c|X>c)\sim X\) |
| Gamma(\(\alpha,\theta\)) | \(\dfrac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\theta^\alpha}\) | See MGF table; use Poisson for integer \(\alpha\) | \(\alpha\theta\) | \(\alpha\theta^2\) | Sum of \(\alpha\) indep. Exp(\(\theta\)) ~ Gamma(\(\alpha,\theta\)) |
| Normal(\(\mu,\sigma^2\)) | \(\dfrac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}\) | \(\Phi\!\left(\dfrac{x-\mu}{\sigma}\right)\) | \(\mu\) | \(\sigma^2\) | Symmetry: \(\Phi(-z)=1-\Phi(z)\) |
| Lognormal(\(\mu,\sigma^2\)) | \(\dfrac{1}{x\sigma\sqrt{2\pi}}e^{-(\ln x-\mu)^2/2\sigma^2}\) | \(\Phi\!\left(\dfrac{\ln x-\mu}{\sigma}\right)\) | \(e^{\mu+\sigma^2/2}\) | \(e^{2\mu+\sigma^2}(e^{\sigma^2}-1)\) | \(\ln X\sim N(\mu,\sigma^2)\) |
| Beta(a,b) | \(\dfrac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}\) | No closed form | \(\dfrac{a}{a+b}\) | \(\dfrac{ab}{(a+b)^2(a+b+1)}\) | Beta(1,1)~Uniform(0,1) |
| Pareto(\(\alpha,\theta\)) | \(\dfrac{\alpha\theta^\alpha}{(x+\theta)^{\alpha+1}}\) | \(1-\left(\dfrac{\theta}{x+\theta}\right)^\alpha\) | \(\dfrac{\theta}{\alpha-1}\) (\(\alpha>1\)) | \(\dfrac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)}\) (\(\alpha>2\)) | Heavy-tailed; used in severity models |
Moment Generating Function Reference5
MGF Definition & Properties
\(M_X(t) = \text{E}[e^{tX}]\)
\(\text{E}[X^k] = M_X^{(k)}(0)\)
\(\text{E}[X] = M'(0),\quad \text{E}[X^2] = M''(0)\)
\(\text{Var}[X] = M''(0) - [M'(0)]^2\)
\(\text{E}[X^k] = M_X^{(k)}(0)\)
\(\text{E}[X] = M'(0),\quad \text{E}[X^2] = M''(0)\)
\(\text{Var}[X] = M''(0) - [M'(0)]^2\)
MGF by Distribution
Binomial: \((1-p+pe^t)^n\)
Poisson: \(e^{\lambda(e^t-1)}\)
Geometric: \(\dfrac{pe^t}{1-(1-p)e^t}\)
Exponential: \((1-\theta t)^{-1}\)
Gamma: \((1-\theta t)^{-\alpha}\)
Normal: \(e^{\mu t+\sigma^2 t^2/2}\)
Poisson: \(e^{\lambda(e^t-1)}\)
Geometric: \(\dfrac{pe^t}{1-(1-p)e^t}\)
Exponential: \((1-\theta t)^{-1}\)
Gamma: \((1-\theta t)^{-\alpha}\)
Normal: \(e^{\mu t+\sigma^2 t^2/2}\)
Gamma CDF via Poisson
For integer \(\alpha\):
\[1 - F_\Gamma(x) = \sum_{k=0}^{\alpha-1}\frac{e^{-x/\theta}(x/\theta)^k}{k!} = \Pr(Y \ge \alpha)\] where \(Y \sim \text{Poisson}(x/\theta)\)
\[1 - F_\Gamma(x) = \sum_{k=0}^{\alpha-1}\frac{e^{-x/\theta}(x/\theta)^k}{k!} = \Pr(Y \ge \alpha)\] where \(Y \sim \text{Poisson}(x/\theta)\)
Multivariate Distributions6
Marginal & Conditional
\(f_X(x) = \displaystyle\int f_{X,Y}(x,y)\,dy\)
\(f_{X|Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)}\)
\(f_{X|Y}(x|y) = \dfrac{f_{X,Y}(x,y)}{f_Y(y)}\)
Covariance & Correlation
\(\text{Cov}[X,Y] = \text{E}[XY] - \text{E}[X]\text{E}[Y]\)
\(\text{Cov}[aX,bY] = ab\,\text{Cov}[X,Y]\)
\(\text{Var}[aX+bY] = a^2\text{Var}[X] + b^2\text{Var}[Y] + 2ab\,\text{Cov}[X,Y]\)
\(\rho_{X,Y} = \dfrac{\text{Cov}[X,Y]}{\sqrt{\text{Var}[X]}\sqrt{\text{Var}[Y]}}\)
\(\text{Cov}[aX,bY] = ab\,\text{Cov}[X,Y]\)
\(\text{Var}[aX+bY] = a^2\text{Var}[X] + b^2\text{Var}[Y] + 2ab\,\text{Cov}[X,Y]\)
\(\rho_{X,Y} = \dfrac{\text{Cov}[X,Y]}{\sqrt{\text{Var}[X]}\sqrt{\text{Var}[Y]}}\)
Double Expectation & Total Variance
\(\text{E}[X] = \text{E}[\text{E}[X|Y]]\)
\(\text{Var}[X] = \text{E}[\text{Var}[X|Y]] + \text{Var}[\text{E}[X|Y]]\)
\(\text{Var}[X] = \text{E}[\text{Var}[X|Y]] + \text{Var}[\text{E}[X|Y]]\)
Independence
\(f_{X,Y}(x,y) = f_X(x)\cdot f_Y(y)\)
\(\text{E}[h(X)\cdot k(Y)] = \text{E}[h(X)]\cdot\text{E}[k(Y)]\)
\(\text{Cov}[X,Y] = 0\)
\(\text{E}[h(X)\cdot k(Y)] = \text{E}[h(X)]\cdot\text{E}[k(Y)]\)
\(\text{Cov}[X,Y] = 0\)
I.I.D. Sums & CLT
\(\text{E}[S_n] = n\mu,\quad \text{Var}[S_n] = n\sigma^2\)
\(\text{E}[\bar{X}] = \mu,\quad \text{Var}[\bar{X}] = \sigma^2/n\)
CLT: \(S_n \approx N(n\mu, n\sigma^2)\) for large \(n\)
\(\text{E}[\bar{X}] = \mu,\quad \text{Var}[\bar{X}] = \sigma^2/n\)
CLT: \(S_n \approx N(n\mu, n\sigma^2)\) for large \(n\)
Order Statistics (i.i.d.)
\(F_{X_{(n)}}(x) = [F_X(x)]^n\) (maximum)
\(S_{X_{(1)}}(x) = [S_X(x)]^n\) (minimum)
\(f_{X_{(k)}}(x) = \dfrac{n!}{(k-1)!(n-k)!}[F]^{k-1}f[S]^{n-k}\)
\(S_{X_{(1)}}(x) = [S_X(x)]^n\) (minimum)
\(f_{X_{(k)}}(x) = \dfrac{n!}{(k-1)!(n-k)!}[F]^{k-1}f[S]^{n-k}\)
Transformations of Random Variables7
CDF Method (univariate)
For \(Y = g(X)\), \(g\) strictly increasing:
\(F_Y(y) = F_X(g^{-1}(y))\)
\(f_Y(y) = f_X(g^{-1}(y))\cdot\left|\dfrac{dx}{dy}\right|\)
\(F_Y(y) = F_X(g^{-1}(y))\)
\(f_Y(y) = f_X(g^{-1}(y))\cdot\left|\dfrac{dx}{dy}\right|\)
Useful Results
If \(X\sim\text{Exp}(\theta)\), then \(2X/\theta\sim\chi^2_2\)
If \(X\sim\text{Gamma}(\alpha,\theta)\), then \(X/\theta\sim\text{Gamma}(\alpha,1)\)
If \(Z\sim N(0,1)\), then \(Z^2\sim\chi^2_1\)
If \(X\sim\text{Uniform}(0,1)\), then \(-\theta\ln X\sim\text{Exp}(\theta)\)
If \(X\sim\text{Gamma}(\alpha,\theta)\), then \(X/\theta\sim\text{Gamma}(\alpha,1)\)
If \(Z\sim N(0,1)\), then \(Z^2\sim\chi^2_1\)
If \(X\sim\text{Uniform}(0,1)\), then \(-\theta\ln X\sim\text{Exp}(\theta)\)
Mixed Distributions
Mixture with weights \(a_1+a_2=1\):
\(F_Y(y) = a_1 F_{C_1}(y) + a_2 F_{C_2}(y)\)
\(\text{E}[Y^k] = a_1\text{E}[C_1^k] + a_2\text{E}[C_2^k]\)
\(F_Y(y) = a_1 F_{C_1}(y) + a_2 F_{C_2}(y)\)
\(\text{E}[Y^k] = a_1\text{E}[C_1^k] + a_2\text{E}[C_2^k]\)
Insurance & Risk Management8
Coverage Modifications -- Payment Y Given Loss X
| Type | Payment Y | E[Y] (general) | E[Y] (exponential) |
|---|---|---|---|
| Deductible d | \(Y = (X-d)_+ = \begin{cases}0 & X\le d \\ X-d & X>d\end{cases}\) | \(\displaystyle\int_d^\infty S_X(x)\,dx\) | \(\theta\cdot\Pr(X>d) = \theta e^{-d/\theta}\) |
| Policy limit u | \(Y = \min(X,u) = \begin{cases}X & X\le u \\ u & X>u\end{cases}\) | \(\displaystyle\int_0^u S_X(x)\,dx\) | \(\theta(1-e^{-u/\theta})\) |
| Deductible d + Limit u | \(Y = \begin{cases}0 & X\le d \\ X-d & d| \(\displaystyle\int_d^{d+u} S_X(x)\,dx\) |
\(\theta(e^{-d/\theta}-e^{-(d+u)/\theta})\) |
|
Unreimbursed loss: \(Z = X - Y\), so \(\text{E}[Z] = \text{E}[X] - \text{E}[Y]\)
Compound Distributions
\(S = X_1 + X_2 + \cdots + X_N\) (N random)
\(\text{E}[S] = \text{E}[N]\cdot\text{E}[X]\)
\(\text{Var}[S] = \text{E}[N]\cdot\text{Var}[X] + \text{Var}[N]\cdot(\text{E}[X])^2\)
For \(N\sim\text{Poisson}(\lambda)\): \(\text{Var}[S] = \lambda\text{E}[X^2]\)
\(\text{E}[S] = \text{E}[N]\cdot\text{E}[X]\)
\(\text{Var}[S] = \text{E}[N]\cdot\text{Var}[X] + \text{Var}[N]\cdot(\text{E}[X])^2\)
For \(N\sim\text{Poisson}(\lambda)\): \(\text{Var}[S] = \lambda\text{E}[X^2]\)
Frequency Severity Models
Stop-loss: \(\text{E}[(S-d)_+] = \text{E}[S] - d + \text{E}[(d-S)_+]\)
Per-occurrence deductible on aggregate \(S\):
\(\text{E}[S_d] = \text{E}[N]\cdot\text{E}[(X-d)_+]\)
Per-occurrence deductible on aggregate \(S\):
\(\text{E}[S_d] = \text{E}[N]\cdot\text{E}[(X-d)_+]\)