IIT JAM MS key terms

The least real number that is greater than or equal to every element of a given set; it may or may not belong to the set itself.

The greatest real number that is less than or equal to every element of a given set; unlike a minimum it need not be a member of the set.

A set of real numbers that fits inside some finite interval, meaning it possesses both a finite upper bound and a finite lower bound.

A sequence whose terms eventually become arbitrarily close to one another; in the real numbers every such sequence converges to a limit.

A sequence of numbers that approaches a single fixed value as the index grows without bound, so its terms can be made arbitrarily close to that value.

The property of the real line that every nonempty set bounded above has a least upper bound, and every Cauchy sequence of reals converges within the reals.

The value that a function's output approaches as its input approaches a specified point, regardless of whether the function is defined at that point.

A property of a function whereby its value at a point equals the limit of its values near that point, so small input changes produce small output changes.

The property that a function has a well-defined derivative at a point, meaning its graph has a nonvertical tangent there; it implies continuity at that point.

A result stating that for a continuous function differentiable on an open interval, some interior point has derivative equal to the function's average rate of change over the interval.

A special case of the mean value theorem: if a function is continuous on a closed interval, differentiable inside it, and equal at both endpoints, some interior point has zero derivative.

A definite integral defined as the limit of sums of rectangle areas over finer and finer partitions of a bounded interval, existing when those sums agree in the limit.

The result linking differentiation and integration: the integral of a derivative recovers the original function, and the derivative of an accumulated area function returns the integrand.

A strong form of convergence for a sequence of functions in which the rate of convergence is the same at every point of the domain.

A map between vector spaces that preserves vector addition and scalar multiplication, sending the zero vector to the zero vector and lines through the origin to lines or points.

The dimension of the column space of a matrix, equal to the number of linearly independent columns and to the number of nonzero rows in its row-reduced form.

A scalar assigned to a square matrix that is nonzero exactly when the matrix is invertible and that measures how the associated linear map scales volumes.

A scalar for which some nonzero vector is mapped to a scalar multiple of itself by a square matrix, that scalar being the multiplier involved.

A nonzero vector whose direction is unchanged by a given square matrix, so applying the matrix merely rescales it by the corresponding eigenvalue.

The condition that a square matrix can be written as a product involving an invertible matrix, a diagonal matrix, and the inverse of that first matrix.

A set with an associative binary operation, an identity element, and an inverse for every element; the operation need not be commutative.

A subset of a group that is itself a group under the same operation, closed under the operation and under taking inverses, and containing the identity.

A group in which every element is a power of one particular element, called a generator; such groups are always commutative.

A rearrangement of the elements of a finite set, viewed as a bijection from the set onto itself and often written in cycle notation.

A function assigning a real number to each outcome of a random experiment, so probabilities of events become probabilities about numerical values.

For a discrete random variable, a function giving the probability that the variable takes each specific value; its values sum to one over the support.

For a continuous random variable, a nonnegative function whose integral over an interval equals the probability that the variable falls in that interval.

A function giving, for each real number, the probability that a random variable is less than or equal to that number; it is nondecreasing and tends to zero and one.

The probability-weighted average value of a random variable, computed as a sum for discrete variables and as an integral for continuous ones.

The expected squared deviation of a random variable from its mean, measuring spread; its square root is the standard deviation.

A measure of how two random variables vary together, defined as the expectation of the product of their deviations from their respective means.

A standardized 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 function defined as the expectation of the exponential of a scaled random variable; its derivatives at zero yield the moments when it exists near zero.

A result stating that sums or averages of many independent, identically distributed variables with finite variance are approximately normally distributed when the sample size is large.

A continuous bell-shaped distribution determined by a mean and a variance, symmetric about its mean and fully specified by those two parameters.

A result stating that the sample average of independent, identically distributed observations converges to the population mean as the sample size grows.

A statistic, computed from sample data, used to approximate an unknown population parameter; it is itself a random variable with its own distribution.

An estimator whose expected value equals the true value of the parameter being estimated, so it is correct on average over repeated sampling.

An estimator that converges in probability to the true parameter value as the sample size increases without bound.

The parameter value that maximizes the likelihood function, namely the joint probability or density of the observed sample treated as a function of the parameter.

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

A random interval computed from sample data that contains an unknown parameter with a specified long-run probability under repeated sampling.

The standard deviation of the sampling distribution of a statistic, quantifying how much the statistic typically varies from sample to sample.

A decision procedure comparing a null hypothesis with an alternative by computing a test statistic and rejecting the null when the statistic falls in a critical region.

The probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one actually observed.

The error of rejecting a true null hypothesis; its probability is the significance level chosen for the test, conventionally written as alpha.

The error of failing to reject a false null hypothesis; its probability is written as beta, and one minus beta is called the power of the test.