Free CAS MAS-I (Modern Actuarial Statistics I) Formula Sheet (2026)

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All MAS-I Formulas

Probability Models 30 items
Gamma waiting time density for the n-th arrival
fSn(s)=λnsn−1e−λs(n−1)!f_{S_n}(s) = \frac{\lambda^n s^{n-1} e^{-\lambda s}}{(n-1)!}
λ = rate, n = event index, s = waiting time, mean n/λ, variance n/λ²
Compound Poisson mean and variance
E[S(t)]=λt E[X], Var(S(t))=λt E[X2]E[S(t)] = \lambda t\,E[X],\ \text{Var}(S(t)) = \lambda t\,E[X^2]
λ = rate, t = time, X = iid severity, E[X²] = Var(X)+(E[X])²
NHPP mean function over an interval
Λ(a,b)=∫abλ(s) ds\Lambda(a,b) = \int_a^b \lambda(s)\,ds
λ(s) = time-varying intensity, [a,b] = interval; equals both mean and variance of N(b)−N(a)
Monte Carlo sample size for target half-width
n≈(1.96 s/h)2n \approx (1.96\,s/h)^2
s = pilot sample SD, h = target half-width, n = required number of draws
Exponential inversion from a uniform
X=−1λln⁡(1−U)X = -\frac{1}{\lambda}\ln(1-U)
U = Uniform(0,1) draw, λ = exponential rate, X = exponential severity draw
Monte Carlo 95% confidence interval half-width
θ^n±1.96 s/n\hat\theta_n \pm 1.96\,s/\sqrt{n}
\hat\theta_n = sample mean of g(X_i), s = sample SD, n = independent draws
Inversion method draw from a uniform
X=F−1(U)X = F^{-1}(U)
U = Uniform(0,1) draw, F^{-1} = generalized inverse CDF, X = draw from target distribution F
Reversionary annuity to y after x dies
aˉy∣x=aˉy−aˉxy\bar a_{y|x} = \bar a_y - \bar a_{xy}
aˉy\bar a_y = single-life annuity on y, aˉxy\bar a_{xy} = joint-life annuity
Joint-life survival under common shock
tpxy=tpx∗⋅tpy∗⋅e−λt{}_tp_{xy} = {}_tp_x^* \cdot {}_tp_y^* \cdot e^{-\lambda t}
tp∗{}_tp^* = private survival, λ\lambda = shared hazard rate, t = time
Survival function from cumulative hazard
S(t)=exp⁡ ⁣(−∫0th(s) ds)=e−H(t)S(t) = \exp\!\left(-\int_0^t h(s)\,ds\right) = e^{-H(t)}
H(t) = cumulative hazard, h(s) = hazard rate, S(t) = survival probability
Last-survivor survival probability (inclusion-exclusion)
tpxy‾=tpx+tpy−tpxy{}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_{xy}
tpx{}_tp_x = prob x survives t, tpxy{}_tp_{xy} = joint survival
Mean residual life at age t
e(t)=∫t∞S(u) duS(t)e(t) = \frac{\int_t^\infty S(u)\,du}{S(t)}
S = survival function, t = current age, e(t) = expected remaining lifetime given survival to t
Joint-life continuous annuity under constant force and interest
aˉxy=1μx+μy+δ\bar a_{xy} = \dfrac{1}{\mu_x + \mu_y + \delta}
μx,μy\mu_x,\mu_y = constant forces of mortality, δ\delta = force of interest
Conditional survival probability (t-p-s)
tps=S(s+t)S(s){_t}p_s = \frac{S(s+t)}{S(s)}
S = survival function, s = current age, t = additional years survived
Hazard rate from density and survival
h(t)=f(t)S(t)=−ddtln⁡S(t)h(t) = \frac{f(t)}{S(t)} = -\frac{d}{dt}\ln S(t)
f = density, S = survival function, h = instantaneous failure rate
Constant-force whole life insurance EPV
Aˉx=μμ+δ\bar A_x = \frac{\mu}{\mu+\delta}
μ = constant force of mortality, δ = constant force of interest
Whole life insurance EPV (discrete)
Ax=∑k=0∞vk+1 k∣qxA_x = \sum_{k=0}^{\infty} v^{k+1}\,{}_{k|}q_x
v = 1/(1+i), k∣qx{}_{k|}q_x = prob of death in year k+1 for life age x
Insurance-annuity fundamental identity (discrete)
Ax=1−d a¨xA_x = 1 - d\,\ddot a_x
d = i/(1+i) effective discount rate, a¨x\ddot a_x = whole life annuity-due EPV, AxA_x = whole life insurance EPV
Whole life annuity-due EPV
a¨x=∑k=0∞vk kpx\ddot a_x = \sum_{k=0}^{\infty} v^{k}\,{}_k p_x
v = 1/(1+i), kpx{}_k p_x = prob life age x survives k years
Waiting time to the n-th event in a Poisson process
Sn=∑i=1nTi∼Gamma(n,λ)S_n = \sum_{i=1}^n T_i \sim \text{Gamma}(n,\lambda), E[Sn]=n/λE[S_n]=n/\lambda
TiT_i = iid Exponential(λ) gaps, n = event number, λ = rate
Non-homogeneous Poisson process mean function
N(b)−N(a)∼Poisson(∫abλ(s) ds)N(b)-N(a) \sim \text{Poisson}\left(\int_a^b \lambda(s)\,ds\right)
λ(s) = intensity function, [a,b] = time interval
Poisson process count probability
P(N(t)=k)=e−λt(λt)kk!P(N(t)=k) = \frac{e^{-\lambda t}(\lambda t)^{k}}{k!}
λ = rate, t = interval length, k = number of events
Fundamental matrix of an absorbing Markov chain
N=(I−Q)−1N = (I - Q)^{-1}
I = identity, Q = transient-to-transient block of P, NijN_{ij} = expected visits to transient state j starting from i
Series system reliability with independent components
Rs=∏i=1nRiR_s = \prod_{i=1}^{n} R_i
RiR_i = reliability of component i, n = number of components in series
Bridge reliability by conditioning on the center element
Rbridge=RC⋅Rworks+(1−RC)⋅RfailsR_{\text{bridge}} = R_C \cdot R_{\text{works}} + (1 - R_C) \cdot R_{\text{fails}}
RCR_C = bridge element reliability, RworksR_{works} = system reliability given C works, RfailsR_{fails} = given C fails
Parallel system reliability with independent components
Rp=1−∏i=1n(1−Ri)R_p = 1 - \prod_{i=1}^{n}(1 - R_i)
RiR_i = reliability of component i, 1 - RiR_i = unreliability, n = number of parallel components
Limited expected value (survival form)
E[X∧u]=∫0uS(x) dxE[X \wedge u] = \int_0^{u} S(x)\,dx
X = non-negative loss, u = cap/limit, S(x) = survival function 1−F(x)
Loss elimination ratio for ordinary deductible
LER(d)=E[X∧d]/E[X]\text{LER}(d) = E[X \wedge d] / E[X]
d = ordinary deductible, X = ground-up loss severity
Expected layer cost between deductible d and limit u
E[min⁡(X,u)−min⁡(X,d)]=E[X∧u]−E[X∧d]E[\min(X,u) - \min(X,d)] = E[X \wedge u] - E[X \wedge d]
X = loss, d = attachment, u = exhaustion point
Exponential limited expected value
E[X∧u]=θ(1−e−u/θ)E[X \wedge u] = \theta(1 - e^{-u/\theta})
θ = exponential mean, u = policy limit/cap
Statistics 35 items
Collective risk model aggregate loss
S=∑i=1NXiS = \sum_{i=1}^{N} X_i
N = random claim count, XiX_i = iid severities independent of N
Compound distribution variance
Var(S)=E[N] Var(X)+Var(N) E[X]2\text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,E[X]^2
N = claim count, X = severity, S = aggregate loss
Panjer recursion for aggregate loss pmf
fS(sh)=11−afX(0)∑y=1s(a+b y/s) fX(yh) fS((s−y)h)f_S(sh) = \frac{1}{1 - a f_X(0)} \sum_{y=1}^{s} (a + b\,y/s)\, f_X(yh)\, f_S((s-y)h)
(a,b) = (a,b,0) class parameters, h = grid step
Normal approximation stop-loss premium
E[(S−d)+]=σ[ϕ(z)−z(1−Φ(z))]E[(S-d)_+] = \sigma[\phi(z) - z(1-\Phi(z))]
z = (d - E[S])/σ, σ = SD of S, φ = standard normal pdf, Φ = cdf
MLE of the exponential rate parameter
λ^=n/∑Xi=1/Xˉ\hat\lambda = n/\sum X_i = 1/\bar X
n = sample size, ΣXiΣX_i = sufficient statistic, X̄ = sample mean
Exponential family canonical density
f(x;θ)=h(x) c(θ)exp⁡(∑j=1kwj(θ) tj(x))f(x;\theta) = h(x)\,c(\theta)\exp\left(\sum_{j=1}^k w_j(\theta)\,t_j(x)\right)
h, c base functions; w natural params; t sufficient kernels
Fisher-Neyman factorization theorem
f(x1,…,xn;θ)=g(T(x),θ)⋅h(x)f(x_1,\ldots,x_n;\theta) = g(T(x),\theta)\cdot h(x)
T = sufficient statistic, g depends on θ through T, h depends only on data
UMVUE of theta for Uniform(0, theta)
θ^UMVUE=n+1nmax⁡Xi\hat\theta_{\text{UMVUE}} = \tfrac{n+1}{n}\max X_i
n = sample size, max XiX_i = largest order statistic (complete sufficient stat)
One-sample z test statistic for a mean
Z=Xˉ−μ0σ/nZ = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}
X-bar = sample mean, μ₀ = hypothesized mean, σ = population SD, n = sample size
Two-sided p-value for a z test
p=2⋅P(∣Z∣≥∣zobs∣)p = 2 \cdot P(|Z| \ge |z_{\text{obs}}|)
zobsz_{obs} = observed test statistic, probability computed under H₀
Likelihood ratio test statistic
−2ln⁡(L0/L1)∼χk2-2\ln(L_0/L_1) \sim \chi^2_k
L₀ = restricted-model likelihood, L₁ = full-model likelihood, k = number of restrictions
Sample size for target power in a one-sided z test
n=σ2(z1−α+z1−β)2(μ0−μ1)2n = \frac{\sigma^2 (z_{1-\alpha} + z_{1-\beta})^2}{(\mu_0 - \mu_1)^2}
σ = SD, α = Type I rate, β = Type II rate, μ₀ = null mean, μ₁ = alternative mean
Nelson-Aalen cumulative hazard estimator
H^(t)=∑tj≤tsj/nj\hat H(t) = \sum_{t_j \le t} s_j/n_j
sⱼ = events at time tⱼ, nⱼ = risk-set size (counts only those with dᵢ < tⱼ ≤ yᵢ)
Conditional density under left-truncation at a deductible
fX∣X>d(x)=f(x)/S(d)f_{X\mid X>d}(x) = f(x)/S(d) for x>dx>d
f = unconditional density, S(d) = survival at deductible d
Likelihood for a right-censored sample
L(θ)=∏i=1nf(yi)δiS(yi)1−δiL(\theta) = \prod_{i=1}^{n} f(y_i)^{\delta_i} S(y_i)^{1-\delta_i}
f = density, S = survival, δᵢ = 1 if uncensored, 0 if right-censored at yᵢ
Likelihood contribution with left truncation and right censoring
Li(θ)=[f(xi∣θ)/S(di∣θ)]δi[S(ui∣θ)/S(di∣θ)]1−δiL_i(\theta) = [f(x_i\mid\theta)/S(d_i\mid\theta)]^{\delta_i}[S(u_i\mid\theta)/S(d_i\mid\theta)]^{1-\delta_i}
δ=1 if observed, d=truncation, u=censoring point
Fisher information for a single observation
I(θ)=E[(∂log⁡f/∂θ)2]=−E[∂2log⁡f/∂θ2]I(\theta) = E\left[\left(\partial \log f/\partial \theta\right)^2\right] = -E\left[\partial^2 \log f/\partial \theta^2\right]
f = density, θ = parameter
Cramer-Rao lower bound for an unbiased estimator
Var(θ^)≥1/[nI(θ)]\text{Var}(\hat\theta) \ge 1/[n I(\theta)]
n = sample size, I(θ) = Fisher information per observation, θ = parameter
Mean squared error decomposition
MSE(θ^)=Var(θ^)+[Bias(θ^)]2\text{MSE}(\hat\theta) = \text{Var}(\hat\theta) + [\text{Bias}(\hat\theta)]^2
Bias(θ̂) = E[θ̂] − θ, Var = sampling variance of the estimator
CDF of the sample minimum
F(1)(x)=1−[1−F(x)]nF_{(1)}(x) = 1 - [1 - F(x)]^{n}
F = parent CDF, n = sample size, X(1)X_{(1)} = minimum
Unbiased sample variance with Bessel's correction
S2=1n−1∑i=1n(Xi−Xˉ)2S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2
n = sample size, XiX_i = i-th observation, Xˉ\bar{X} = sample mean, S² unbiased for σ²
Computational shortcut for sum of squared deviations
∑i=1n(Xi−Xˉ)2=∑i=1nXi2−nXˉ2\sum_{i=1}^{n}(X_i-\bar{X})^2 = \sum_{i=1}^{n} X_i^2 - n\bar{X}^2
n = sample size, XiX_i = i-th observation, Xˉ\bar{X} = sample mean
Density of the k-th order statistic
f(k)(x)=n!(k−1)!(n−k)!F(x)k−1[1−F(x)]n−kf(x)f_{(k)}(x) = \frac{n!}{(k-1)!(n-k)!} F(x)^{k-1}[1-F(x)]^{n-k} f(x)
F = CDF, f = pdf, n = sample size, k = rank
CDF of the sample maximum
F(n)(x)=[F(x)]nF_{(n)}(x) = [F(x)]^{n}
F = parent CDF, n = sample size, X(n)X_{(n)} = maximum
Standard error of the sample mean
SE(Xˉ)=S/n\text{SE}(\bar{X}) = S/\sqrt{n}
S = sample standard deviation, n = sample size; uses σ/√n when σ known
Sample mean
Xˉ=1n∑i=1nXi\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i
n = sample size, XiX_i = i-th observation, Xˉ\bar{X} = sample mean (unbiased for μ)
Uniform order statistic as a Beta distribution
U(k)∼Beta(k, n−k+1)U_{(k)} \sim \text{Beta}(k,\, n-k+1), E[U(k)]=k/(n+1)E[U_{(k)}] = k/(n+1)
n = sample size, k = rank, U = Uniform(0,1) sample
Z-test statistic for a single mean with known variance
Z=Xˉ−μ0σ/nZ = \dfrac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}
Xˉ\bar{X} = sample mean, μ0\mu_0 = hypothesized mean, σ\sigma = known population SD, n = sample size
T-test statistic for a single mean with unknown variance
T=Xˉ−μ0s/n∼tn−1T = \dfrac{\bar{X} - \mu_0}{s/\sqrt{n}} \sim t_{n-1}
Xˉ\bar{X} = sample mean, μ0\mu_0 = hypothesized mean, s = sample SD, n = sample size
Exact for an iid normal sample under H0H_0. Without normality it holds only approximately, for large n.
F-test statistic for the ratio of two variances
F=s12s22∼Fn1−1, n2−1F = \dfrac{s_1^2}{s_2^2} \sim F_{n_1-1,\,n_2-1}
s12,s22s_1^2, s_2^2 = sample variances (larger on top), n1,n2n_1, n_2 = sample sizes
Requires two independent iid normal samples and equal variances under H0H_0. Not valid for non-normal data, even for large samples.
Chi-square test statistic for a single variance
W=(n−1)s2σ02∼χn−12W = \dfrac{(n-1)s^2}{\sigma_0^2} \sim \chi^2_{n-1}
n = sample size, s2s^2 = sample variance, σ02\sigma_0^2 = hypothesized variance
Requires an iid normal sample under H0H_0. Not valid for non-normal data, even for large n.
Lognormal mean and variance
E[X]=eμ+σ2/2, Var(X)=e2μ+σ2(eσ2−1)E[X] = e^{\mu+\sigma^2/2},\ \text{Var}(X) = e^{2\mu+\sigma^2}(e^{\sigma^2}-1)
μ, σ = mean and SD of log X
Aggregate loss mean and variance (general two-term)
E[S]=E[N] E[X], Var(S)=E[N] Var(X)+Var(N) E[X]2E[S] = E[N]\,E[X],\ \text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,E[X]^2
N = claim count, X = iid severity
Negative binomial mean and variance
E[N]=rβ, Var(N)=rβ(1+β)E[N] = r\beta,\ \text{Var}(N) = r\beta(1+\beta)
r = shape, β = scale; variance exceeds mean by factor (1+β)
(a,b,0) class recursion
pk/pk−1=a+b/kp_k/p_{k-1} = a + b/k
pkp_k = probability of k claims, a and b = family-specific constants (Poisson, NegBin, Binomial)
Extended Linear Models 35 items
Pearson chi-square dispersion estimate
ϕ^=X2/(n−p)\hat\phi = X^2/(n-p) where X2=∑(riP)2X^2 = \sum (r_i^P)^2
n = sample size, p = parameter count, rir_i^P = Pearson residual
Deviance of a GLM
D=2[ℓ(y;y)−ℓ(μ^;y)]D = 2[\ell(\mathbf{y};\mathbf{y}) - \ell(\hat{\boldsymbol{\mu}};\mathbf{y})]
ℓ(y;y) = saturated log-likelihood, ℓ(μ̂;y) = fitted log-likelihood
Pearson residual for a GLM
riP=(yi−μ^i)/V(μ^i)r_i^{P} = (y_i - \hat\mu_i)/\sqrt{V(\hat\mu_i)}
yiy_i = observed, μ̂_i = fitted mean, V(μ̂_i) = variance function at μ̂_i
McFadden pseudo R-squared
RMcF2=1−ℓmodel/ℓnullR^2_{\text{McF}} = 1 - \ell_{\text{model}}/\ell_{\text{null}}
ℓ_model = fitted log-likelihood, ℓ_null = intercept-only log-likelihood
Incremental pure-premium GLM with base-rate offset
ln⁡E[Pi]=ln⁡(Bi)+β0+∑jβjxij\ln E[P_i] = \ln(B_i) + \beta_0 + \sum_j \beta_j x_{ij}
P = pure premium, B = current base premium, β = log-relativities to the base
Annualized Poisson claim frequency from an exposure-offset model
λ^i=μi/Ei=exp⁡(β0+∑jβjxij)\hat\lambda_i = \mu_i / E_i = \exp(\beta_0 + \sum_j \beta_j x_{ij})
λ = per-exposure rate, μ = expected count, E = earned exposure
Population-averaged prediction across a control variable
μˉ=∑kpk g−1(ηk)\bar\mu = \sum_k p_k \, g^{-1}(\eta_k)
pkp_k = population share of control level k, ηkη_k = linear predictor at level k, g = link function
Linear predictor in a GLM with an offset term
ηi=oi+β0+∑jβjxij\eta_i = o_i + \beta_0 + \sum_j \beta_j x_{ij}
o = known offset (coef fixed at 1), β = estimated coefficients, x = predictors, η = linear predictor
Score equation under the canonical link
X⊤(y−μ)=0X^\top (y - \mu) = 0
X = design matrix, y = response vector, μ = fitted mean vector at the MLE
GLM response variance with dispersion and exposure weight
Var(Yi)=ϕ V(μi)/wi\mathrm{Var}(Y_i) = \phi\, V(\mu_i)/w_i
φ = dispersion, V = variance function, μ = mean, w = exposure weight
Log-link GLM with exposure offset
ln⁡μi=ln⁡(exposurei)+xi⊤β\ln \mu_i = \ln(\text{exposure}_i) + x_i^\top \beta
μ = mean response, exposure = policy-years at risk, x = covariates, β = coefficients
Exponential family density form
f(y;θ,ϕ)=exp⁡{(yθ−b(θ))/a(ϕ)+c(y,ϕ)}f(y;\theta,\phi) = \exp\{(y\theta - b(\theta))/a(\phi) + c(y,\phi)\}
θ = canonical parameter, φ = dispersion, b = cumulant function, a,c = known functions
Likelihood ratio statistic for nested GLMs
Λ=2(ℓ1−ℓ0)∼˙χΔp2\Lambda = 2(\ell_1 - \ell_0) \dot\sim \chi^2_{\Delta p}
ℓ₁ = full-model log-likelihood, ℓ₀ = reduced-model log-likelihood, Δp = extra parameters
Akaike information criterion for GLM selection
AIC=−2ℓ+2p\text{AIC} = -2\ell + 2p
ℓ = maximized log-likelihood, p = number of fitted parameters; lower is better
Elastic net penalized GLM objective
β^=arg⁡min⁡β{−ℓ(β)+λ[α∥β∥1+(1−α)∥β∥22]}\hat\beta = \arg\min_\beta \{-\ell(\beta) + \lambda[\alpha\|\beta\|_1 + (1-\alpha)\|\beta\|_2^2]\}
λ = penalty strength, α = L1/L2 mix (1 = lasso, 0 = ridge)
Bayesian information criterion for GLM selection
BIC=−2ℓ+pln⁡n\text{BIC} = -2\ell + p\ln n
ℓ = maximized log-likelihood, p = parameters, n = sample size; lower is better
Extended linear model linear predictor and link
η=β0+∑j=1pβjxj,  g(μ)=η\eta = \beta_0 + \sum_{j=1}^{p} \beta_j x_j,\; g(\mu)=\eta
η = linear predictor, β = coefficients, x = design columns, g = link, μ = mean response
Continuous-by-continuous interaction model
η=β0+β1x1+β2x2+β3(x1x2)\eta = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 (x_1 x_2)
x1, x2 = continuous predictors, β3 = interaction coefficient measuring departure from additivity
Degrees of freedom for a categorical-by-categorical interaction
dfint=(k1−1)(k2−1)df_{int} = (k_1 - 1)(k_2 - 1)
k1, k2 = number of levels in the two categorical predictors; added on top of (k1-1)+(k2-1) main-effect df
Effective slope on x1 under a continuous interaction
∂η/∂x1=β1+β3x2\partial \eta / \partial x_1 = \beta_1 + \beta_3 x_2
β1 = main-effect slope, β3 = interaction coefficient, x2 = partner predictor value
Standardized residual
ri=eis1−hiir_i = \dfrac{e_i}{s\sqrt{1-h_{ii}}}
eie_i = raw residual, s = residual std error, hiih_{ii} = leverage of point i
Added variable plot residuals for predictor X_j
ey∣−j=y−y^(−j), ej∣−j=Xj−X^j,(−j)e_{y|-j} = y - \hat{y}_{(-j)},\ e_{j|-j} = X_j - \hat{X}_{j,(-j)}
hats with (-j) = fitted values from regressions that exclude XjX_j
Hat matrix for ordinary least squares
H=X(XTX)−1XTH = X(X^TX)^{-1}X^T
X = design matrix; hiih_{ii} = i-th diagonal of H is the leverage of observation i
Cook's distance for an observation
Di=ri2p+1⋅hii1−hiiD_i = \dfrac{r_i^2}{p+1} \cdot \dfrac{h_{ii}}{1-h_{ii}}
rir_i = standardized residual, hiih_{ii} = leverage, p = number of predictors
Deviance-based pseudo R-squared for a GLM
Rdev2=1−DmodelDnullR^2_{\text{dev}} = 1 - \frac{D_{\text{model}}}{D_{\text{null}}}
DmodelD_{model} = deviance of fitted GLM, DnullD_{null} = deviance of intercept-only model
Adjusted R-squared for linear regression
Radj2=1−(1−R2) n−1n−k−1R^2_{\text{adj}} = 1 - (1-R^2)\,\frac{n-1}{n-k-1}
n = sample size, k = number of slope parameters (intercept excluded)
Coefficient of determination for OLS with intercept
R2=1−SSESST=SSRSSTR^2 = 1 - \frac{\text{SSE}}{\text{SST}} = \frac{\text{SSR}}{\text{SST}}
SSE = error sum of squares, SST = total sum of squares, SSR = regression sum of squares
Scaled deviance of a GLM from log-likelihoods
D∗=2(ℓsat−ℓmodel)D^* = 2(\ell_{\text{sat}} - \ell_{\text{model}}), with unscaled D=ϕD∗D = \phi D^*
ℓ_sat = saturated log-likelihood, ℓ_model = fitted log-likelihood, φ = dispersion
Freedman-Diaconis rule for histogram bin width
h=2⋅IQR/n1/3h = 2 \cdot IQR / n^{1/3}
h = bin width, IQR = interquartile range of the data, n = sample size
Interquartile range
IQR=Q3−Q1IQR = Q_3 - Q_1
Q1 = first quartile (25th percentile), Q3 = third quartile (75th percentile)
Tukey upper outlier fence for a box plot
Upper fence=Q3+1.5⋅IQR\text{Upper fence} = Q_3 + 1.5 \cdot IQR
Q3 = third quartile, IQR = interquartile range; points above are flagged outliers
Tukey lower outlier fence for a box plot
Lower fence=Q1−1.5⋅IQR\text{Lower fence} = Q_1 - 1.5 \cdot IQR
Q1 = first quartile, IQR = interquartile range; points below are flagged outliers
F statistic for analysis of deviance with estimated dispersion
F=(ΔD/Δdf)/ϕ^F = (\Delta D / \Delta df) / \hat\phi
ΔD = deviance reduction from added terms, Δdf = added parameters, φ̂ = estimated dispersion
Wald z statistic for a GLM coefficient
z=β^j/SE(β^j)z = \hat\beta_j / \text{SE}(\hat\beta_j)
β̂_j = MLE of coefficient j, SE = standard error of the estimate
Pearson chi-square goodness-of-fit statistic for a GLM
X2=∑(yi−μ^i)2/V(μ^i)X^2 = \sum (y_i - \hat\mu_i)^2 / V(\hat\mu_i)
yiy_i = observation, μ̂_i = fitted mean, V = variance function
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Frequently Asked Questions

Is the MAS-I formula sheet free?
Yes. The full MAS-I formula sheet is free, with no signup, no email, and no credit card required. 100 formulas across 3 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.
Can I download the MAS-I formula sheet as a printable PDF?
Yes. A 1080x1350 portrait PDF (Instagram and LinkedIn carousel native size, also great for tablet study) is linked at the top of this page. The PDF is fully self-contained: math is pre-rendered, fonts are embedded, no internet connection needed once downloaded.
What's covered on the MAS-I formula sheet?
Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.
What is FreeFellow's relationship with CAS?
No. FreeFellow is not affiliated with the CAS or any examination body. This is an independent study aid covering the published syllabus.
What else is free at FreeFellow for MAS-I candidates?
The full original question bank is free with an account, subject to usage limits. Worked solutions, written lessons, mixed practice, and your readiness score stay free. The formula sheet is free too. Fellow is $39 per month or $79 per quarter, per exam family (USD). Fellow Plus is $49 per month, $99 per quarter, or $199 per year, per exam family (USD). Every annual plan is Fellow Plus. Fellow adds timed mock exams, spaced-repetition flashcards, performance analytics, and a personalized study plan.

Formula sheets for other exams

Exam P Exam FM Exam FAM Exam SRM Exam PA Exam ALTAM Exam ASTAM CFA Level I CFA Level II CFA L3 Portfolio Mgmt CFA L3 Private Wealth CFA L3 Private Markets

About FreeFellow

Jeffrey Ting, founder of FreeFellow
Jeffrey Ting
FSA, CFA · Founder

FreeFellow was built by Jeffrey Ting, a credentialed actuary and CFA charterholder who has passed thirteen of the hardest exams in finance, all on his first attempt. He paid four-figure prep fees along the way.

So he started writing his own questions, then lessons, then mock exams, until it grew into a full prep platform covering 40 exams with more than 45,000 original practice questions.

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A CFA charter, a CPA license, an actuarial credential. Each opens a real career, but prep fees add to the cost of getting there. FreeFellow keeps its original question bank and written lessons free so you can study even if a paid course is out of reach.

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The original question bank, the worked solutions, the topic lessons, mixed practice, and your readiness score stay free, forever. Full access needs a free account, and practice usage limits apply. Fellow adds AI grading on supported exams, spaced-repetition flashcards, full-length mock exams, analytics, and a study plan that adapts to your progress.

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