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[1] [2] Also dating from the latter half of the 19th century, the inequality attributed to Chebyshev described bounds on a distribution when only the mean and variance of the variable are known, and the related inequality attributed to Markov found bounds on a positive variable when only the mean is known.
Given a sample from a normal distribution, whose parameters are unknown, it is possible to give prediction intervals in the frequentist sense, i.e., an interval [a, b] based on statistics of the sample such that on repeated experiments, X n+1 falls in the interval the desired percentage of the time; one may call these "predictive confidence intervals".
[6] [7] It is also known as Fréchet-Cramér–Rao or Fréchet-Darmois-Cramér-Rao lower bound. It states that the precision of any unbiased estimator is at most the Fisher information; or (equivalently) the reciprocal of the Fisher information is a lower bound on its variance.
Chebyshev's inequality requires the following information on a random variable : . The expected value [] is finite.; The variance [] = [( [])] is finite.; Then, for every constant >,
Confidence bands can be constructed around estimates of the empirical distribution function.Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods.
The probability density function (PDF) for the Wilson score interval, plus PDF s at interval bounds. Tail areas are equal. Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval ( , + ) has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test.
Upper and lower probabilities are representations of imprecise probability. Whereas probability theory uses a single number, the probability , to describe how likely an event is to occur, this method uses two numbers: the upper probability of the event and the lower probability of the event.
Decision boundaries can be approximations of optimal stopping boundaries. [ 2 ] The decision boundary is the set of points of that hyperplane that pass through zero. [ 3 ] For example, the angle between a vector and points in a set must be zero for points that are on or close to the decision boundary.