Free resource

CS1 formula cheat sheet

The statistics CS1 keeps testing, on one page: what each formula says, when it applies, and the plain Memori syntax that types it. The grey text under each name doubles as a reference for writing your own cards.

Distributions

The standard families and the moments the tables questions assume you know cold.

Binomial moments

Variance below the mean: the opposite tell to overdispersed counts

E[X] = np, Var[X] = np(1-p)

E[X] = np, Var[X] = np(1-p)

Poisson probability function

Mean and variance are both λ

P(X = x) = (e^{-lambda} lambda^x)/(x!)

P(X = x) = e-λ λxx!

Exponential density

Memoryless; mean 1/λ

f(x) = lambda e^{-lambda x}

f(x) = λ e-λ x

Gamma moments

Chi-squared is the α = ν/2, λ = 1/2 special case

E[X] = (alpha)/(lambda), Var[X] = (alpha)/(lambda^2)

E[X] = αλ, Var[X] = αλ2

Lognormal mean

μ, σ² are the moments of ln X, not of X: the classic trap

E[X] = exp(mu + (sigma^2)/(2))

E[X] = exp(μ + σ22)

Standardising a normal

Probabilities then come from the Φ tables

Z = (X - mu)/(sigma)

Z = X - μσ

Moments and dependence

Moment generating function

Differentiate r times at t = 0 for the r-th moment

M_X(t) = E[e^{tX}]

M_X(t) = E[etX]

Variance shortcut

Var(X) = E[X^2] - E[X]^2

Var(X) = E[X2] - E[X]2

Covariance shortcut

Zero under independence, but not conversely

Cov(X,Y) = E[XY] - E[X]E[Y]

Cov(X,Y) = E[XY] - E[X]E[Y]

Correlation coefficient

Linear association only, always in [−1, 1]

rho = (Cov(X,Y))/(sqrt(Var(X)Var(Y)))

ρ = Cov(X,Y)√(Var(X)Var(Y))

Tower formulas

The engine behind compound distributions and credibility

E[X] = E[E[X|Y]], Var(X) = E[Var(X|Y)] + Var(E[X|Y])

E[X] = E[E[X|Y]], Var(X) = E[Var(X|Y)] + Var(E[X|Y])

Sampling

Central limit theorem

Whatever the parent, given a finite variance

(bar(X) - mu)/(sigma / sqrt(n)) -> N(0, 1)

X - μσ / √(n) → N(0, 1)

Sample variance

n − 1 makes it unbiased for σ²

S^2 = (1)/(n-1) sum_{i=1}^{n} (X_i - bar(X))^2

S2 = 1n-1 n∑i=1 (X_i - X)2

t statistic

Replaces z when σ is estimated

t = (bar(X) - mu)/(S / sqrt(n)) ~ t_{n-1}

t = X - μS / √(n) ~ t_n-1

Variance sampling distribution

Needs normal data; invert for a CI on σ²

((n-1)S^2)/(sigma^2) ~ chi^2_{n-1}

(n-1)S2σ2 ~ χ2n-1

Two-sample variance ratio

F with n₁ − 1 and n₂ − 1 degrees of freedom under equality

F = (S_1^2)/(S_2^2)

F = S21S22

Estimation and testing

Maximum likelihood equation

Log first; check the second derivative is negative

(d)/(d theta) ln L(theta) = 0

dd θ ln L(θ) = 0

Cramér-Rao lower bound

The MLE attains it asymptotically

CRLB = (1)/(I(theta)), I(theta) = -E[(d^2)/(d theta^2) ln L]

CRLB = 1I(θ), I(θ) = -E[d2d θ2 ln L]

Mean square error decomposition

A little bias can buy a big variance saving

MSE = Var + Bias^2

MSE = Var + Bias2

CI for a mean, σ known

z = 1.96 for 95%; width shrinks with √n

bar(X) pm z_{alpha/2} (sigma)/(sqrt(n))

X ± z_α/2 σ√(n)

Regression

Least squares slope

The fitted line always passes through (x̄, ȳ)

b = (S_{xy})/(S_{xx})

b = S_xyS_xx

Intercept

a = bar(y) - b bar(x)

a = y - b x

Coefficient of determination

The squared sample correlation between x and y

R^2 = (S_{xy}^2)/(S_{xx}S_{yy})

R2 = S2xyS_xxS_yy

Bayesian methods and credibility

Bayes' theorem for densities

Posterior ∝ likelihood × prior

f(theta|x) = (f(x|theta) f(theta))/(f(x))

f(θ|x) = f(x|θ) f(θ)f(x)

Estimator under quadratic loss

Median for absolute loss, mode for 0/1 loss

theta* = E[theta | x]

θ× = E[θ | x]

Credibility premium

Own experience against the collateral estimate

P = Z bar(X) + (1 - Z) mu

P = Z X + (1 - Z) μ

EBCT Model 1 credibility factor

Between-risk variance pushes Z towards the data

Z = (n)/(n + E[s^2(theta)] / Var[m(theta)])

Z = nn + E[s2(θ)] / Var[m(θ)]

Generalised linear models

GLM form

Canonical links: identity, log, logit

g(mu_i) = eta_i = x_i^T beta

g(μ_i) = η_i = xTi β

Scaled deviance

Nested-model differences compared to chi-squared

D = 2(ln L_{sat} - ln L_{mod})

D = 2(ln L_sat - ln L_mod)

See them move. The sampling results live in the CLT demo and statistical tables, credibility has its own calculator, and every distribution here is on the distributions cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CS1 set in the shop. Join the beta.