CS1 formula sheet

S.MONK · CS1 formula sheet · 58 formulae · smonk.in/formula-sheet/cs1

Moment generating functions, and how to get moments out of them

M_X(t) = E[e^tX]

M(t) = ½ e⁰ + ½ e^t = ½ (1 + e^t)

M(t) = E[1 + tX + t²X²/2! + t³X³/3! + …]
= 1 + t E[X] + t² E[X²]/2! + t³ E[X³]/3! + …

E[X^k] = M^(k)(0)

K_X(t) = ln M_X(t)

K′(0) = E[X]
K″(0) = Var(X)
K′″(0) = E[(X − μ)³]

M_X+Y(t) = M_X(t) × M_Y(t)    and    K_X+Y(t) = K_X(t) + K_Y(t)

The central limit theorem, and when the normal approximation is safe

Z_n = (X̄ − μ) / (σ/√n)  →  N(0, 1)

X̄  ≈  N(μ, σ²/n)    for the average
S_n = ∑X_i  ≈  N(nμ, nσ²)    for the total

sd(X̄) = σ√n / n = σ/√n

Maximum likelihood estimation, step by step

L(θ) = f(x₁, …, x_n | θ) = ∏_i f(x_i | θ)    for independent observations

L(p) = p³ (1 − p)⁷
ℓ(p) = 3 ln p + 7 ln(1 − p)
ℓ′(p) = 3/p − 7/(1 − p)

I(θ) = −E[d²ℓ/dθ²]    and    θ̂ ≈ N(θ, 1/I(θ))

Hypothesis testing and p-values, without the folklore

p = P(a test statistic at least as extreme as the one observed | H₀ is true)

Simple linear regression: least squares, fitted by hand

Y_i = α + βx_i + ε_i,    ε_i ~ N(0, σ²) independently

S(α, β) = ∑ (y_i − α − βx_i)²

β̂ = S_xy / S_xx     α̂ = ȳ − β̂ x̄

S_xx = ∑(x_i − x̄)²    S_yy = ∑(y_i − ȳ)²    S_xy = ∑(x_i − x̄)(y_i − ȳ)

S_xy = ∑x_iy_i − (∑x_i)(∑y_i)/n     S_xx = ∑x_i² − (∑x_i)²/n

S_yy = β̂S_xy + SS_res
(total) = (explained by the regression) + (residual)

R² = 1 − SS_res/S_yy = explained / total

σ̂² = SS_res / (n − 2)

se(β̂) = σ̂ / √S_xx     t = β̂ / se(β̂) ~ t_n−2 under H₀: β = 0

Generalised linear models: the three parts, and the link function

η = β₀ + β₁x₁ + … + β_px_p

g(μ) = η,    where μ = E[Y]

μ = e^{β₀ + β₁x₁ + β₂x₂} = e^β₀ × e^β₁x₁ × e^β₂x₂

ln(μ) = ln(exposure) + β₀ + β₁x₁ + …

D = 2[ℓ_saturated − ℓ_model]

D = 2 ∑ [o ln(o/e) − (o − e)]

Bayesian estimation, conjugate priors and the credibility formula inside them

π(θ | x) = f(x | θ) π(θ) / f(x)   →   posterior ∝ likelihood × prior

posterior mean = (α + ∑x)/(β + n)

posterior mean = Z × x̄ + (1 − Z) × (α/β),    where Z = n/(n + β)

Conditional expectation, and the variance decomposition that comes with it

E[X] = E[ E[X | Y] ]

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

Sampling distributions: what the sample mean and sample variance actually do

E[X̄] = μ     Var(X̄) = σ²/n

S² = Σ(X_i − X̄)² / (n − 1)    gives    E[S²] = σ²

The method of moments, and when maximum likelihood is worth the extra work

E[X] = x̄     E[X²] = Σx_i²/n     E[X³] = Σx_i³/n    and so on

1/λ = x̄   →   λ̂ = 1/x̄

m₂ = Σ(x_i − x̄)²/n     s² = Σ(x_i − x̄)²/(n − 1)

The likelihood ratio test, and why the Wald and score tests exist too

Λ = (largest likelihood allowed under H₀) / (largest likelihood allowed overall)

−2 ln Λ = 2[ ℓ(θ̂) − ℓ(θ₀) ]

W = (θ̂ − θ₀)² I(θ̂)

S = [ℓ′(θ₀)]² / I(θ₀)

The chi-square goodness of fit test, and how to get its degrees of freedom right

χ² = Σ (O_i − E_i)² / E_i

df = (number of categories) − 1 − (number of parameters estimated from the data)

Contingency tables: testing whether two classifications are related

E_ij = n × (R_i/n) × (C_j/n) = R_i C_j / n

χ² = ΣΣ (O_ij − E_ij)² / E_ij

df = (r − 1)(c − 1)

df = rc − 1 − (r − 1) − (c − 1) = rc − r − c + 1 = (r − 1)(c − 1)

Analysis of variance: comparing three or more means at once

0.95³ = 0.857375   →   P(at least one false alarm) = 0.142625

ΣΣ(x_ij − x̄)²  =  Σ n_j(x̄_j − x̄)²  +  ΣΣ(x_ij − x̄_j)²

SS_total = SS_between + SS_within

MS_between = SS_between/(k − 1)     MS_within = SS_within/(N − k)
F = MS_between / MS_within

Multiple linear regression, and what controlling for another variable really does

y_i = α + β₁x_1i + β₂x_2i + … + ε_i

α̂ = ȳ − β̂₁x̄₁ − β̂₂x̄₂

S_yy = SS_regression + SS_residual

s² = SS_residual/(n − p − 1)

R²_adj = 1 − (1 − R²)(n − 1)/(n − p − 1)

Free from S.MONK School of Actuaries · smonk.in · every formula here is explained, with a worked example, on the page it links to.