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Applications - Exam P formula sheet Every Exam P formula in applications on one printable sheet, condensed from the 5 reference pages that teach them.
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Applications CDF method F Y ( y ) = P ( g ( X ) ≤ y ) F_Y(y) = P(g(X)\le y) F Y ( y ) = P ( g ( X ) ≤ y ) then differentiate Monotone transform f Y ( y ) = f X ( g − 1 ( y ) ) ∣ d d y g − 1 ( y ) ∣ f_Y(y) = f_X\big(g^{-1}(y)\big)\left|\dfrac{d}{dy}g^{-1}(y)\right| f Y ( y ) = f X ( g − 1 ( y ) ) d y d g − 1 ( y ) Maximum of n iid F max ( x ) = [ F ( x ) ] n F_{\max}(x) = \left[F(x)\right]^{n} F m a x ( x ) = [ F ( x ) ] n Minimum of n iid S min ( x ) = [ S ( x ) ] n S_{\min}(x) = \left[S(x)\right]^{n} S m i n ( x ) = [ S ( x ) ] n Convolution f X + Y ( s ) = ∫ f X ( x ) f Y ( s − x ) d x f_{X+Y}(s) = \int f_X(x)f_Y(s-x)\,dx f X + Y ( s ) = ∫ f X ( x ) f Y ( s − x ) d x Ordinary deductible E [ ( X − d ) + ] = ∫ d ∞ S ( x ) d x E[(X-d)_{+}] = \int_d^{\infty} S(x)\,dx E [( X − d ) + ] = ∫ d ∞ S ( x ) d x Policy limit E [ X ∧ u ] = ∫ 0 u S ( x ) d x E[X \wedge u] = \int_0^{u} S(x)\,dx E [ X ∧ u ] = ∫ 0 u S ( x ) d x Decomposition E [ X ] = E [ X ∧ d ] + E [ ( X − d ) + ] E[X] = E[X\wedge d] + E[(X-d)_{+}] E [ X ] = E [ X ∧ d ] + E [( X − d ) + ] Exponential deductible E [ ( X − d ) + ] = e − λ d λ E[(X-d)_{+}] = \dfrac{e^{-\lambda d}}{\lambda} E [( X − d ) + ] = λ e − λ d memorylessness Coinsurance E [ α ( X − d ) + ] = α E [ ( X − d ) + ] E[\alpha(X-d)_{+}] = \alpha E[(X-d)_{+}] E [ α ( X − d ) + ] = α E [( X − d ) + ] Sum moments E [ ∑ X i ] = n μ , Var ( ∑ X i ) = n σ 2 E\!\left[\sum X_i\right] = n\mu, \qquad \operatorname{Var}\!\left(\sum X_i\right) = n\sigma^{2} E [ ∑ X i ] = n μ , Var ( ∑ X i ) = n σ 2 Standardisation Z = ∑ X i − n μ σ n → d N ( 0 , 1 ) Z = \dfrac{\sum X_i - n\mu}{\sigma\sqrt{n}} \xrightarrow{d} N(0,1) Z = σ n ∑ X i − n μ d N ( 0 , 1 ) Sample mean X ˉ ≈ N ( μ , σ 2 n ) \bar{X} \approx N\!\left(\mu, \dfrac{\sigma^{2}}{n}\right) X ˉ ≈ N ( μ , n σ 2 ) Continuity correction P ( S ≥ k ) ≈ 1 − Φ ( k − 0.5 − n μ σ n ) P(S \ge k) \approx 1 - \Phi\!\left(\dfrac{k - 0.5 - n\mu}{\sigma\sqrt{n}}\right) P ( S ≥ k ) ≈ 1 − Φ ( σ n k − 0.5 − n μ ) Multinomial probability P ( N 1 = k 1 , … , N m = k m ) = n ! k 1 ! ⋯ k m ! p 1 k 1 ⋯ p m k m P(N_1=k_1,\dots,N_m=k_m) = \dfrac{n!}{k_1!\cdots k_m!}\,p_1^{k_1}\cdots p_m^{k_m} P ( N 1 = k 1 , … , N m = k m ) = k 1 ! ⋯ k m ! n ! p 1 k 1 ⋯ p m k m Marginals N j ∼ Binomial ( n , p j ) N_j \sim \text{Binomial}(n, p_j) N j ∼ Binomial ( n , p j ) Covariance Cov ( N i , N j ) = − n p i p j , i ≠ j \operatorname{Cov}(N_i, N_j) = -n p_i p_j, \quad i \ne j Cov ( N i , N j ) = − n p i p j , i = j Constraint ∑ j k j = n , ∑ j p j = 1 \sum_j k_j = n, \qquad \sum_j p_j = 1 ∑ j k j = n , ∑ j p j = 1 Mixed density f ( x ) = ∑ k w k f k ( x ) , ∑ k w k = 1 f(x) = \sum_k w_k f_k(x), \qquad \sum_k w_k = 1 f ( x ) = ∑ k w k f k ( x ) , ∑ k w k = 1 Mixed mean E [ X ] = ∑ k w k μ k E[X] = \sum_k w_k \mu_k E [ X ] = ∑ k w k μ k Mixed second moment E [ X 2 ] = ∑ k w k ( σ k 2 + μ k 2 ) E[X^{2}] = \sum_k w_k\left(\sigma_k^{2} + \mu_k^{2}\right) E [ X 2 ] = ∑ k w k ( σ k 2 + μ k 2 ) Variance Var ( X ) = ∑ k w k ( σ k 2 + μ k 2 ) − ( ∑ k w k μ k ) 2 \operatorname{Var}(X) = \sum_k w_k\left(\sigma_k^{2}+\mu_k^{2}\right) - \left(\sum_k w_k\mu_k\right)^{2} Var ( X ) = ∑ k w k ( σ k 2 + μ k 2 ) − ( ∑ k w k μ k ) 2 Applications - Exam P formula sheet. Formulas condensed from the free reference pages on this site. Not affiliated with or endorsed by the Society of Actuaries.