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Free SOA Exam PA Formula Sheet PDF, No Signup (2026)

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All Exam PA Formulas

Predictive Analytics Problem Definition 7 items

Profit per policy

Profit = P - L - E

, P = premium (price) per policy, L = expected loss (claim severity times frequency), E = expense or acquisition cost per policy

Value as price minus cost

V = P - C

, V = value of the item, P = price (e.g. sale price per building), C = cost (e.g. construction cost per building)

Base rate of a binary target

p = \frac{n _{e v e n t}}{N}

, p = base rate (positive-class proportion), n_event = number of positive-class records, N = total records

Majority-class (no-information) accuracy

a c c_{maj} = 1 - p

, acc_maj = accuracy of always predicting the majority class, p = base rate of the rare (positive) class

Business impact of a predictive model

Impact = V \times f

, V = dollar value of the decision the prediction improves, f = how often that decision is made

Additive error data-generating model

y = f (x) + ε

, y = observed response, f(x) = true signal function,

ε

= random noise with variance

Var (ε)

Bias-variance decomposition of expected test MSE

E [(y_{0} - f^(x_{0}))^{2}] = Var (f^(x_{0})) + [Bias (f^(x_{0}))]^{2} + Var (ε)

f^

= fitted model,

x_{0}

= test point,

y_{0}

= true response,

Var (ε)

= irreducible error

Data Exploration and Visualization 8 items

Variance of a factor-level estimate

Var (μ^_{level}) \propto \frac{1}{n _{level}}

μ^_{level}

= estimated mean for that level,

n_{level}

= number of rows in that level

Minority rows needed to oversample to a target share

f = \frac{pN}{1 - p}

from

p = \frac{f}{N + f}

, p = target minority proportion, N = majority-class row count, f = minority rows after oversampling

Boxplot outlier fences

lower = Q 1 - 1.5 \times IQR, upper = Q 3 + 1.5 \times IQR

, Q1 = first quartile, Q3 = third quartile, IQR = interquartile range

Interquartile range

IQR = Q 3 - Q 1

, Q1 = 25th percentile (first quartile), Q3 = 75th percentile (third quartile)

Shared variance between two predictors

r^{2} = r_{x y}^{2}

r_{x y}

= Pearson correlation between the two variables;

r^{2}

= proportion of one variable's variation explained by the other

Sample variance

s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - x ˉ)^{2}

x_{i}

= observation,

x ˉ

= sample mean, n = sample size

Pearson correlation coefficient

r_{x y} = \frac{\sum ( x _{i} - x ˉ ) ( y _{i} - y ˉ )}{\sum ( x _{i} - x ˉ ) ^{2} \sum ( y _{i} - y ˉ ) ^{2}}

, x,y = paired values,

x ˉ, y ˉ

= means, i = observation index

Sample standard deviation

s = s^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - x ˉ)^{2}

s^{2}

= sample variance,

x_{i}

= observation,

x ˉ

= mean, n = sample size

Data Transformations and Unsupervised Learning Techniques 12 items

Standardized feature (z-score for clustering)

z = \frac{x - μ}{σ}

, x = raw feature value, μ = feature mean, σ = feature standard deviation

Complete linkage between-cluster distance

d (A, B) = max_{a \in A, b \in B} ∥ a - b ∥

, A, B = clusters, a, b = member points, distance = farthest pair

Proportion of variance explained by a principal component

PVE_{m} = \frac{Var ( z _{m} )}{\sum _{j} Var ( z _{j} )}

, PVEm = proportion for PC m, Var(zm) = variance of PC m, denominator = total variance across all PCs

Variance of a principal component from its standard deviation

Var (z_{m}) = s_{m}^{2}

, Var(zm) = variance captured by PC m, sm = reported standard deviation of PC m; total variance = number of standardized variables

Within-cluster sum of squares (K-means objective)

WCSS = \sum_{k = 1}^{K} \sum_{x \in C_{k}} ∥ x - μ_{k} ∥^{2}

, K = number of clusters, C_k = cluster k, x = observation, μ_k = mean vector of cluster k

Single linkage between-cluster distance

d (A, B) = min_{a \in A, b \in B} ∥ a - b ∥

, A, B = clusters, a, b = member points, distance = closest pair

First principal component score

z_{1} = ϕ_{11} x_{1} + ϕ_{21} x_{2} + ϕ_{31} x_{3}

, z1 = PC1 score, φj1 = loading of variable j on PC1, xj = standardized feature j

Loading vector normalization constraint

\sum_{j} ϕ_{j m}^{2} = 1

, φjm = loading of variable j on PC m; squared loadings of a component sum to one (unit-length loading vector)

Number of dummy columns for a categorical variable

d = k - 1

, k = number of category levels, d = number of dummy (indicator) columns needed to avoid redundancy in a GLM

Log transform for a skewed non-negative variable

x^{'} = ln (x + 1)

, x = raw strictly non-negative value, x' = transformed value; the +1 lets x = 0 be handled

Min-max scaling

x_{mm} = \frac{x - x _{m i n}}{x _{m a x} - x _{m i n}}

, x = raw value,

x_{m i n}

= minimum,

x_{m a x}

= maximum,

x_{mm}

= scaled value in [0,1]

Z-score standardization

z = \frac{x - x ˉ}{s}

, x = raw value,

x ˉ

= sample mean, s = sample standard deviation, z = standardized value

Generalized Linear Models 15 items

Poisson log-link model with a log-exposure offset

lo g (E [count]) = lo g (exposure) + β_{0} + β_{1} x_{1} + \dots

, exposure = units of exposure (offset, coef fixed at 1),

β_{0}

= intercept,

β_{j}

= slope,

x_{j}

= predictor

Weighted least squares objective

min_{β} \sum_{i} w_{i} (y_{i} - y^_{i})^{2}

w_{i}

= weight of row i,

y_{i}

= observed response,

y^_{i}

= fitted value,

β

= coefficients

Group-size weighted mean of grouped rates

y ˉ_{w} = \frac{\sum _{i} n _{i} y _{i}}{\sum _{i} n _{i}}

n_{i}

= group size (weight) of row i,

y_{i}

= group-average rate for row i

Expected count as exposure times a fitted rate

E [count] = exposure \times e^{β_{0} + β_{1} x_{1}}

, exposure = units of exposure,

e^{β_{0} + β_{1} x_{1}}

= rate per unit exposure,

β_{0}

= intercept,

β_{1}

= slope,

x_{1}

= predictor

GLM log link function

lo g (μ) = β_{0} + β_{1} x_{1} + \dots

, so

e^{β_{1}}

is the factor by which a one-unit rise in

x_{1}

scales

μ

;

μ

= expected response,

β

= coefficients

Pearson residual for a GLM

r_{i} = \frac{y _{i} - μ ^ _{i}}{V ( μ ^ _{i} )}

y_{i}

= observed,

μ^_{i}

= fitted mean,

V (μ^_{i})

= variance function at the fitted mean

Lognormal mean back-transformation (smearing correction)

E^[Y] = e^{μ^_{l o g}} e^{σ^{2} /2}

μ^_{l o g}

= fitted value on log scale,

σ^{2}

= residual variance on log scale

Estimated dispersion parameter

ϕ^= \frac{1}{n - p} \sum_{i = 1}^{n} \frac{( y _{i} - μ ^ _{i} ) ^{2}}{V ( μ ^ _{i} )}

, n = observations, p = parameters,

y_{i}

= observed,

μ^_{i}

= fitted mean,

V

= variance function

Elastic net penalized regression objective

min_{β} \frac{1}{2 n} \sum_{i} (y_{i} - β_{0} - \sum_{j} β_{j} x_{ij})^{2} + λ [\frac{1 - α}{2} \sum_{j} β_{j}^{2} + α \sum_{j} ∣ β_{j} ∣]

, λ = penalty strength, α = L1/L2 mix, β = coefficients

Variance-inflation factor for a predictor

VIF_{j} = \frac{1}{1 - R _{j}^{2}}

, R_j^2 = R-squared from regressing predictor j on all other predictors

One-standard-error threshold for lambda.1se selection

threshold = MSE_{m i n} + SE_{m i n}

, MSE_min = minimum CV mean squared error, SE_min = its standard error; lambda.1se = largest λ with CV MSE ≤ threshold

Log-link multiplicative effect of a coefficient

μ_{n e w} = μ \times e^{β}

per unit, μ = expected response, β = coefficient on the link scale;

e^{β} - 1

is the percent change

Logit-link odds ratio from a coefficient

O R = e^{β}

, OR = multiplicative change in the odds of the event per unit increase, β = logit-link coefficient

Interaction-adjusted slope of a predictor at a level

slope of x_{1} at level ℓ = β_{1} + β_{int, ℓ}

, β1 = main coefficient, β_int,ℓ = interaction coefficient (zero for baseline level)

GLM linear predictor with link function

g (μ) = η = β_{0} + β_{1} x_{1} + \dots

, g = link function, μ = expected response, η = linear predictor, β = coefficients, x = predictors

Tree-Based Models 11 items

Default mtry for random forest classification

m_{t r y} = p

, p = total number of predictors, m_try = predictors sampled at each split

Out-of-bag probability a row is left out of a bootstrap sample

P = (1 - 1/ n)^{n} \to e^{- 1} \approx 0.368

, n = number of training rows, P = chance a given row is omitted from one bootstrap (~37%)

Default mtry for random forest regression

m_{t r y} = p /3

, p = total number of predictors, m_try = predictors sampled at each split

Gradient boosting additive update

F_{m} (x) = F_{m - 1} (x) + λ h_{m} (x)

, F = ensemble prediction, m = boosting round, λ = learning rate (shrinkage), h_m = tree fit to current pseudo-residuals

k-fold cross-validation error

CV error = \frac{1}{k} \sum_{i = 1}^{k} Error (fold_{i})

, k = number of folds, fold_i = i-th held-out validation fold

Variance of an averaged tree ensemble

Var = ρ σ^{2} + \frac{1 - ρ}{B} σ^{2}

, ρ = pairwise tree correlation, σ² = single-tree prediction variance, B = number of trees

Learning-rate to tree-count inverse tradeoff

n_{n e w} \approx n_{o l d} \times \frac{λ _{o l d}}{λ _{n e w}}

, n = number of trees, λ = learning rate; halving λ roughly doubles the trees needed

Cost-complexity pruning criterion

R_{c p} (T) = R (T) + c p \cdot ∣ T ∣

, R(T) = tree total error, cp = complexity parameter, |T| = number of terminal nodes (leaves)

Gini impurity of a node

G = 1 - \sum_{k} p_{k}^{2}

, G = Gini impurity, p_k = proportion of class k in the node; G = 0 at a pure node (rpart default)

Entropy of a node

H = - \sum_{k} p_{k} lo g_{2} p_{k}

, H = entropy in bits, p_k = proportion of class k in the node; H = 0 at a pure node

Root mean square error

RMSE = \frac{1}{n} \sum_{i} (y_{i} - y^_{i})^{2}

, y_i = actual value,

y^_{i}

= predicted value, n = number of observations

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