Formula sheets

Bayes' theorem $P(A_i\mid B)=\frac{P(B\mid A_i)P(A_i)}{\sum_j P(B\mid A_j)P(A_j)}$

Conditional variance (law of total variance) $\operatorname{Var}(X)=E[\operatorname{Var}(X\mid Y)]+\operatorname{Var}(E[X\mid Y])$

MGF moments $\;E[X^k]=M_X^{(k)}(0)$; sum of independent variables: $M_{X+Y}(t)=M_X(t)M_Y(t)$.

Expected payment with deductible $d$ on loss $X$ (per loss): $E[(X-d)_+]=\int_d^\infty S_X(x)\,dx$.

Exponential memorylessness $\;P(X>s+t\mid X>s)=P(X>t)$; for $X\sim\text{Exp}(\theta)$, $E[(X-d)_+]=\theta e^{-d/\theta}$.

Order statistics for i.i.d. sample of size $n$: $F_{(n)}(x)=F(x)^n$, $F_{(1)}(x)=1-(1-F(x))^n$.