IFoA CS1 cheat sheet

CS1 spans probability distributions, stochastic processes, survival models, Markov chains, and generalised linear models for actuarial applications.

Discrete distributions: Binomial(n,p) mean np, variance np(1-p); Poisson(lambda) mean=variance=lambda; Geometric(p) counts failures before first success with mean (1-p)/p.

Continuous distributions: Exponential(lambda) has mean 1/lambda, variance 1/lambda^2, memoryless property; Gamma(alpha,lambda) is the sum of alpha independent Exponential(lambda) variables when alpha is an integer.

Normal approximation to the binomial is valid when np and n(1-p) are both large; apply a continuity correction of 0.5.

Moment generating function M_X(t)=E[e^{tX}]; it uniquely determines the distribution and derivatives give moments: M^(r)(0)=E[X^r].

Central limit theorem: for i.i.d. variables with mean mu and variance sigma^2, the standardised sample mean converges to N(0,1) as n grows.

Law of total probability: P(A)=sum over i of P(A|B_i)P(B_i) for a partition {B_i}; Bayes' theorem reverses conditioning: P(B_j|A)=P(A|B_j)P(B_j)/P(A).

Covariance Cov(X,Y)=E[XY]-E[X]E[Y]; Var(aX+bY)=a^2Var(X)+b^2Var(Y)+2abCov(X,Y), with independence implying zero covariance but not conversely.

Survival function S(x)=P(T>x)=1-F(x); it is non-increasing with S(0)=1, and the force of mortality is mu(x)=-d/dx ln S(x).

Relations: S(x)=exp(-integral from 0 to x of mu(t)dt); f(x)=S(x)mu(x); the curtate future lifetime K has P(K=k)=S(k)-S(k+1).

Complete expectation of life e_x = integral from 0 to infinity of S(x+t)dt divided by S(x); curtate expectation e_x = sum over k>=1 of S(x+k)/S(x).

Actuarial present value of a whole life insurance paying 1 at death: A_x = integral of v^t * f(t) dt over t>=0, where v=1/(1+i) is the discount factor.

Whole life annuity-due a_x = sum over k>=0 of v^k * kP_x; relationship a_x = (1-A_x)/d where d=i/(1+i) is the discount rate.

Under the constant force of mortality assumption with mu, S(t)=exp(-mu*t) and A_x = mu/(mu+delta) where delta=ln(1+i).

Fractional age assumptions: uniform distribution of deaths (UDD) gives tP_x = 1 - t*q_x for 0<=t<=1; constant force gives tP_x = (p_x)^t.

Life table columns: l_x, d_x=l_x-l_{x+1}, q_x=d_x/l_x, p_x=l_{x+1}/l_x, and L_x is the person-years lived between ages x and x+1.

A Markov chain satisfies the memoryless property P(X_{n+1}=j | X_0,...,X_n=i)=P(X_{n+1}=j | X_n=i)=p_ij.

The n-step transition matrix is P^(n) = P^n (matrix power of the one-step transition matrix); the Chapman-Kolmogorov equations express p_ij^(m+n)=sum over k of p_ik^(m) p_kj^(n).

A state i is recurrent if return is certain and transient otherwise; state i is periodic with period d if returns are only possible in multiples of d.

Stationary distribution pi satisfies pi*P=pi with sum of pi_i=1; for an irreducible, aperiodic, positive recurrent chain, pi_i = 1/mu_i where mu_i is the mean return time.

A Poisson process with rate lambda has inter-arrival times that are i.i.d. Exponential(lambda) and counts N(t) ~ Poisson(lambda*t); it has independent and stationary increments.

In a birth-death process, the stationary distribution satisfies pi_j = pi_0 * product from k=1 to j of (lambda_{k-1}/mu_k), normalised so probabilities sum to 1.

Expected hitting times solve linear equations: h_i = 1 + sum over j of p_ij*h_j for transient states, with h_i=0 for absorbing target states.

The gambler's ruin result for a simple random walk with absorbing barriers at 0 and N gives probability of reaching N from i as i/N when the walk is symmetric.

A GLM specifies a random component (exponential family), a systematic component eta=X*beta, and a link g such that g(E[Y])=eta.

Canonical links: identity for Normal, log for Poisson, logit for Binomial, and inverse for Gamma; canonical links simplify the score equations.

Exponential family form: f(y;theta,phi)=exp{(y*theta - b(theta))/a(phi) + c(y,phi)}; E[Y]=b'(theta) and Var(Y)=a(phi)*b''(theta).

The variance function V(mu) characterises each family: Normal V=1, Poisson V=mu, Binomial V=mu(1-mu), Gamma V=mu^2.

The deviance is D=2*(log-likelihood of saturated model minus log-likelihood of fitted model); it is used to compare nested models via chi-squared approximation.

Maximum likelihood estimates solve score equations; the Fisher information matrix I(beta) has entries -E[d^2 l / d beta_i d beta_j], and Var(beta_hat) is approximately I^(-1).

For a Poisson GLM with log link, coefficients are interpreted multiplicatively: a one-unit increase in x_k multiplies the fitted mean by exp(beta_k).

Model checking uses Pearson or deviance residuals; overdispersion occurs when the residual deviance greatly exceeds its degrees of freedom, suggesting a quasi-likelihood or negative binomial alternative.