IFoA CS1 flashcards

31 cards. Read the term and the definition here. The deck at the top of this page is optional.

Survival function

S(x) equals the probability that a lifetime exceeds x, is non-increasing, equals one at age zero, and relates to the force of mortality by S(x)=exp(-integral of mu from 0 to x).

Force of mortality

mu(x) is the instantaneous hazard of death at age x given survival to that age; it equals f(x)/S(x) and also the negative derivative of ln S(x).

Curtate future lifetime

The integer K counting complete future years lived before death. Its probability function is P(K=k)=kP_x*q_{x+k}, and its expectation e_x sums survival probabilities over future years.

Whole life insurance APV

A_x is the expected present value of one unit paid at the moment of death, computed as the integral of v^t times the density of future lifetime over all positive t.

Annuity-due relationship

The whole life annuity-due a_x equals (1-A_x)/d, where d=i/(1+i) is the discount rate; this links insurance and annuity present values under the same interest and mortality basis.

Uniform distribution of deaths

A fractional age assumption stating that deaths are spread evenly across each year of age, giving tP_x=1-t*q_x for t between zero and one, and linear interpolation of l_x.

Constant force assumption

An assumption that mu is constant within each year of age, so tP_x equals (p_x)^t for t between zero and one, producing exponential rather than linear survival between integer ages.

Life table l_x

The number of survivors to exact age x from a radix cohort; ratios l_{x+t}/l_x give survival probabilities, and differences d_x=l_x-l_{x+1} count deaths in the year.

Markov property

A process has the Markov property when the future depends only on the present state, not on the path taken to reach it; formally the conditional distribution given the whole history equals that given the current state.

Transition matrix

A square matrix P whose entry p_ij is the probability of moving from state i to state j in one step; every row sums to one because probabilities are non-negative.

Chapman-Kolmogorov equations

These state that n-step transition probabilities are entries of the matrix power P^n, equivalently p_ij^(m+n) equals the sum over intermediate states k of p_ik^(m) times p_kj^(n).

Stationary distribution

A probability vector pi satisfying pi*P=pi, meaning the state distribution is unchanged after one transition; it is unique for an irreducible, aperiodic, positive recurrent chain.

Recurrent state

A state is recurrent when the probability of ever returning to it, starting from it, equals one; if that probability is strictly less than one the state is transient.

Periodic state

A state has period d greater than one when returns are possible only after a number of steps that is a multiple of d; period one states are called aperiodic.

Poisson process

A counting process with independent stationary increments where N(t) is Poisson with mean lambda*t; inter-arrival times are independent Exponential(lambda) random variables.

Birth-death process

A continuous-time Markov chain on non-negative integers where transitions only move up by one (birth rate lambda_i) or down by one (death rate mu_i), used to model queues and populations.

Gambler's ruin probability

For a symmetric random walk on states 0 to N with absorbing barriers, the probability of reaching N before 0 starting from i equals i divided by N.

Absorbing state

A state with p_ii=1, so once entered it is never left; expected hitting times of absorbing states are obtained by solving linear equations over the transient states.

Binomial distribution

The count of successes in n independent trials each with success probability p; its mean is np and its variance is np(1-p), and its probabilities use binomial coefficients.

Poisson distribution

A count distribution with probability function exp(-lambda)*lambda^k/k! for non-negative integers k; its mean and variance both equal lambda, and it models rare events in fixed intervals.

Exponential distribution

A continuous distribution with density lambda*exp(-lambda*x) for positive x, mean 1/lambda and variance 1/lambda squared; it is memoryless, so the past gives no information about remaining time.

Moment generating function

M_X(t)=E[exp(tX)]; when it exists in a neighbourhood of zero it uniquely identifies the distribution, and its r-th derivative at zero gives the r-th moment.

Central limit theorem

For independent identically distributed variables with finite mean and variance, the standardised sum converges in distribution to standard normal as the sample size grows, enabling approximate inference.

Bayes' theorem

P(B_j|A) equals P(A|B_j)P(B_j) divided by the sum over all partition members of P(A|B_i)P(B_i); it updates prior probabilities into posterior probabilities after observing A.

Variance of a sum

Var(aX+bY) equals a squared times Var(X) plus b squared times Var(Y) plus twice ab times Cov(X,Y); the covariance term vanishes only when X and Y are uncorrelated.

Generalised linear model

A model combining a response from an exponential family, a linear predictor X*beta, and a link function g with g(E[Y])=X*beta, fitted by maximum likelihood.

Canonical link

The link that equates the linear predictor to the natural parameter theta of the exponential family, such as logit for binomial and log for Poisson responses.

Variance function

The function V(mu) giving Var(Y) as phi times V(mu); it equals mu for Poisson, mu(1-mu) for binomial, and mu squared for gamma families.

Deviance

Twice the difference between the maximised log-likelihood of the saturated model and that of the fitted model; nested models are compared using differences in deviance.

Fisher information

The expected negative second derivative of the log-likelihood with respect to parameters; its inverse approximates the covariance matrix of maximum likelihood estimates for large samples.

Overdispersion

A situation where observed variance exceeds the variance assumed by the model, common for Poisson counts; residual deviance far above its degrees of freedom signals it, and quasi-likelihood may be used.