IAI CS1 key terms

A numerical quantity whose value is determined by the outcome of a random experiment, so each possible value has an associated probability. It may be discrete or continuous.

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

For a continuous random variable, a non-negative function whose integral over any interval equals the probability of falling in that interval. It integrates to one over the whole real line.

The function giving, for each value x, the probability that a random variable is less than or equal to x. It is non-decreasing, right-continuous, and tends to zero and one at the extremes.

The probability-weighted average of a random variable's possible values, computed as a sum for discrete variables or an integral for continuous ones. It is a measure of central location.

The expected squared deviation of a random variable from its mean, measuring spread. It equals the mean of the square minus the square of the mean.

The positive square root of the variance, expressing dispersion in the same units as the original variable. It is often easier to interpret than the variance.

A function defined as the expected value of the exponential of t times the random variable, which, when it exists, uniquely determines the distribution and generates its moments by differentiation.

For a non-negative integer-valued random variable, the expected value of s raised to the power of the variable. Its derivatives at one yield factorial moments.

The probability that a random variable exceeds a given value, equal to one minus the cumulative distribution function. It is widely used for lifetimes and time-to-event data.

The instantaneous rate of failure at a given time conditional on survival up to that time, obtained as the density divided by the survival function. It describes the age-specific risk of failure.

The integral of the hazard function from zero to a given time, equal to minus the logarithm of the survival function. It accumulates the total risk experienced up to that time.

The distribution of the number of successes in a fixed number of independent trials, each with the same probability of success. Its mean is n times p and variance n times p times one minus p.

The distribution of the number of events occurring in a fixed interval when events happen independently at a constant average rate. Its mean and variance are both equal to the rate parameter.

A continuous distribution used to model the waiting time until the first event of a Poisson process. It has a constant hazard and is memoryless.

A symmetric bell-shaped continuous distribution defined by a mean and a variance. Sums of many independent variables tend towards it, which underlies large-sample approximations.

The normal distribution with mean zero and variance one, whose cumulative probabilities are tabulated. Any normal variable is converted to it by subtracting the mean and dividing by the standard deviation.

A result stating that, under mild conditions, the sum or mean of a large number of independent, identically distributed variables is approximately normally distributed, whatever the underlying distribution.

A rule expressing the probability of an event as a weighted average of its conditional probabilities given a set of mutually exclusive and exhaustive cases, weighted by the probabilities of those cases.

A rule for reversing conditional probabilities, expressing the probability of a hypothesis given data in terms of the probability of the data given the hypothesis and the prior probability.

The probability that one event occurs given that another event has occurred, defined as the joint probability of both divided by the probability of the conditioning event, provided that is positive.

A property of two events or variables where knowing one gives no information about the other, so their joint probability or joint distribution factorises into the product of the individual ones.

A measure of the degree to which two random variables vary together, defined as the expected product of their deviations from their respective means. It can be positive, negative, or zero.

A standardised measure of linear association between two variables, obtained by dividing their covariance by the product of their standard deviations. It lies between minus one and one.

A joint distribution for two variables where every linear combination is normally distributed. It is fully specified by the two means, the two variances, and the correlation.

A rule or statistic computed from a sample that is used to estimate an unknown population parameter. It is itself a random variable with a sampling distribution.

The difference between the expected value of an estimator and the true value of the parameter it estimates. An estimator with zero bias for all parameter values is called unbiased.

The expected squared difference between an estimator and the true parameter value. It decomposes into the variance of the estimator plus the square of its bias.

The standard deviation of an estimator's sampling distribution, quantifying how much the estimate would vary across repeated samples. It typically decreases as sample size grows.

The value of a parameter that maximises the likelihood, the probability or density of the observed sample viewed as a function of the parameter. It is often asymptotically efficient and normally distributed.

The natural logarithm of the likelihood function. Maximising it gives the same estimate as maximising the likelihood, and its derivatives are used to obtain variances of estimators.

A measure of how much information a sample carries about a parameter, given by the expected value of the negative second derivative of the log-likelihood. Its reciprocal bounds estimator variance.

A lower limit on the variance of any unbiased estimator of a parameter, equal to the reciprocal of the Fisher information. Estimators achieving it are called efficient.

A range of values constructed from sample data so that, across repeated samples, a stated proportion of such intervals would contain the true parameter value.

A function of the sample data and the unknown parameter whose distribution does not depend on that parameter. It is used to construct confidence intervals and hypothesis tests.

A symmetric continuous distribution with heavier tails than the normal, arising when a normal mean is standardised using an estimated standard deviation. It tends to the normal as degrees of freedom increase.

A procedure for deciding between competing statements about a parameter using sample data, by comparing an observed test statistic with a critical region or by computing a p-value.

The probability, chosen in advance, of rejecting a true null hypothesis. It fixes the size of the critical region and is commonly set at five or one percent.

The probability of correctly rejecting a false null hypothesis, equal to one minus the probability of a Type II error. It depends on the true parameter value and the sample size.