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

Transformations

CDF method: $F_{Y} (y) = P (g (X) \leq y)$ then differentiate
Monotone transform: $f_{Y} (y) = f_{X} (g^{- 1} (y)) \frac{d}{d y} g^{- 1} (y)$
Maximum of n iid: $F_{m a x} (x) = [F (x)]^{n}$
Minimum of n iid: $S_{m i n} (x) = [S (x)]^{n}$
Convolution: $f_{X + Y} (s) = \int f_{X} (x) f_{Y} (s - x) d x$

Deductibles and limits

Ordinary deductible: $E [(X - d)_{+}] = \int_{d}^{\infty} S (x) d x$
Policy limit: $E [X \land u] = \int_{0}^{u} S (x) d x$
Decomposition: $E [X] = E [X \land d] + E [(X - d)_{+}]$
Exponential deductible: $E [(X - d)_{+}] = \frac{e ^{- λ d}}{λ}$ memorylessness
Coinsurance: $E [α (X - d)_{+}] = α E [(X - d)_{+}]$

Central limit theorem

Sum moments: $E [\sum X_{i}] = n μ, Var (\sum X_{i}) = n σ^{2}$
Standardisation: $Z = \frac{\sum X _{i} - n μ}{σ n} d N (0, 1)$
Sample mean: $\overset{ˉ}{X} \approx N (μ, \frac{σ ^{2}}{n})$
Continuity correction: $P (S \geq k) \approx 1 - Φ (\frac{k - 0.5 - n μ}{σ n})$

Repeated trials

Multinomial probability: $P (N_{1} = k_{1}, \dots, N_{m} = k_{m}) = \frac{n !}{k _{1} ! \dots k _{m} !} p_{1}^{k_{1}} \dots p_{m}^{k_{m}}$
Marginals: $N_{j} \sim Binomial (n, p_{j})$
Covariance: $Cov (N_{i}, N_{j}) = - n p_{i} p_{j}, i \neq = j$
Constraint: $\sum_{j} k_{j} = n, \sum_{j} p_{j} = 1$

Mixtures

Mixed density: $f (x) = \sum_{k} w_{k} f_{k} (x), \sum_{k} w_{k} = 1$
Mixed mean: $E [X] = \sum_{k} w_{k} μ_{k}$
Mixed second moment: $E [X^{2}] = \sum_{k} w_{k} (σ_{k}^{2} + μ_{k}^{2})$
Variance: $Var (X) = \sum_{k} w_{k} (σ_{k}^{2} + μ_{k}^{2}) - (\sum_{k} w_{k} μ_{k})^{2}$

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.

Applications - Exam P formula sheet

What is on this sheet

Applications

Applications

Transformations

Deductibles and limits

Central limit theorem

Repeated trials

Mixtures