IFoA CS1 key terms

A function that assigns a numerical value to each outcome in a sample space, so that probabilities can be attached to numbers rather than raw outcomes.

For a discrete random variable, a function giving the probability that the variable equals each possible value; the values are non-negative and sum to one.

For a continuous random variable, a non-negative function whose integral over an interval gives the probability of falling in that interval; the total integral is one.

A function giving, for any value x, the probability that a random variable is less than or equal to x; it is non-decreasing and runs from zero to one.

The probability that a random variable exceeds a given value, equal to one minus the cumulative distribution function; central to lifetime and mortality modelling.

The instantaneous rate of failure at a given age conditional on survival to that age, obtained as the density divided by the survival function.

In life-contingent work, the instantaneous death rate at exact age x conditional on survival to that age; the actuarial name for the hazard rate of a lifetime variable.

The probability-weighted average of a random variable over its distribution; for a function of the variable, the average of that function's values weighted by probability or density.

The expected squared deviation of a random variable from its mean, measuring spread; computed as the second moment minus the square of the first moment.

The expected value of the exponential of t times a random variable, viewed as a function of t; it generates moments by differentiation and can identify distributions uniquely.

For a non-negative integer-valued random variable, the expected value of z raised to the variable; its derivatives at one give factorial moments and it aids convolution of counts.

A measure of joint linear association between two random variables, being the expected product of their deviations from their means; zero does not imply independence.

Covariance standardised by the product of the two standard deviations, giving a value between minus one and one that is invariant to changes of scale.

Variables whose joint distribution factorises into the product of their marginal distributions; equivalently, conditioning on one leaves the distribution of the other unchanged.

The probability law describing two or more random variables simultaneously, from which marginal and conditional distributions can be derived by summing or integrating.

The distribution of one variable obtained from a joint distribution by summing or integrating out the other variables, ignoring them entirely.

The distribution of one variable given that another takes a specified value, formed from the joint distribution divided by the relevant marginal probability or density.

The expected value of one variable given information about another; it is itself a random variable when the conditioning variable is random, and obeys the tower property.

A decomposition stating that the variance of a variable equals the variance of its conditional expectation plus the expectation of its conditional variance.

The result that the standardised sum or mean of many independent, identically distributed variables with finite variance tends to a standard normal distribution as the count grows.

A continuous symmetric bell-shaped distribution defined by a mean and variance, widely used as a large-sample approximation and closed under linear combinations of independent variables.

A discrete distribution for counts with a single rate parameter, where the mean and variance are equal; it arises as a limit of binomial counts and in Poisson processes.

A broad class of distributions whose densities can be written in a standard form with a natural parameter and sufficient statistic, giving convenient estimation and moment properties.

A model for events occurring randomly in time or space where counts in disjoint intervals are independent Poisson variables with rate proportional to interval length.

A distribution for a random sum of independent identically distributed amounts, where the number of terms is itself random; used for aggregate claim models.

The values of a sample arranged in increasing order; their distributions depend on the underlying distribution and are used for extremes and quantile estimation.

A rule or statistic computed from sample data that produces a guess for an unknown parameter; it is a random variable before the data are observed.

The difference between the expected value of an estimator and the true parameter value; an unbiased estimator has zero bias over repeated sampling.

The expected squared difference between an estimator and the true parameter; it decomposes into the square of the bias plus the variance of the estimator.

A property of an estimator meaning it converges in probability to the true parameter value as the sample size grows without limit.

The parameter value that maximises the likelihood, or equivalently the log-likelihood, of the observed sample; often found by solving score equations.

The joint probability or density of the observed data regarded as a function of the unknown parameters, with the data held fixed; it is not a probability distribution over parameters.

The natural logarithm of the likelihood function; maximised at the same point but usually easier to differentiate because products become sums.

The derivative of the log-likelihood with respect to the parameter; its expectation is zero and setting it to zero gives likelihood equations for the estimator.

A measure of how much a sample carries about a parameter, equal to the variance of the score and to the negative expected second derivative of the log-likelihood.

A lower limit on the variance of any unbiased estimator, equal to the reciprocal of the Fisher information for a single parameter.

A comparison of estimators by variance, where the efficient estimator attains the smallest variance among unbiased estimators, often the Cramer-Rao bound.

A statistic that captures all information in the sample about the parameter, so the conditional distribution of the data given it does not depend on the parameter.

A random interval constructed so that, over repeated samples, it contains the true parameter with a specified long-run probability called the confidence level.

A decision procedure comparing data with a null hypothesis using a test statistic, a significance level, and a rejection region or p-value.

The probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as that observed; small values suggest evidence against the null.

The probability of correctly rejecting a false null hypothesis; it increases with effect size, sample size, and the chosen significance level.