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FACTOR ANALYSIS. A branch of multivariate analysis in which the observed variates xi (i = 1,2,...,p) are supposed to be expressible in terms of a number m < p factors fj together with residual elements.
FAILURE. The termination of the ability of any item to perform its required function under stated environmental conditions for a specified period of time. 
FAILURE ANALYSIS. The logical, systematic examination of an item or its diagram(s) to identify and analyze the probability, causes, and consequence of potential and real failures. 
FAILURE, CATASTROPHIC. Failures which are both sudden and complete. 
FAILURE, COMPLETE. Failure resulting from deviations in characteristic(s) beyond specified limits such as to cause complete lack of the required function. Note: The limits referred to in this category are special limits specified for this purpose. 
FAILURE CRITERIA. Rules for failure relevancy such as specified limits for the acceptability of an item. 
FAILURE, DEGRADATION. Failures which are both gradual and partial. 
FAILURE, DEPENDENT. One which is caused by the failure of an associated item(s). Not independent. 
FAILURE, GRADUAL. Failures that could be anticipated by prior examination. 
FAILURE, INDEPENDENT. One which occurs without being related to the failure of associated items. Not dependent. 
FAILURE, INHERENT WEAKNESS. Failures attributable to weakness inherent in the item itself when subjected to stresses within the stated capabilities of that item. 
FAILURE MECHANISM. The physical, chemical or other process which results in a failure. 
FAILURE, MISUSE. Failures attributable to the application of stresses beyond the stated capabilities of the item. 
FAILURE MODE. The effect by which a failure is observed; for example, an open or short circuit condition, or a gain change. 
FAILURE, PARTIAL. Failures resulting from deviations in characteristic(s) beyond specified limits but not such as to cause complete lack of the required function. 
FAILURE, RANDOM. Any failure whose occurrence is unpredictable in an absolute sense but which is predictable only in a probabilistic or statistical sense. 
FAILURE RATE. The rate at which failures occur in a certain time interval; i.e., the probability that a failure per unit time occurs in the interval, given that a failure has not occured prior to the start of the interval. 
FAILURE RATE ACCELERATION FACTOR. The ratio of the accelerated testing failure rate understated reference test conditions and time period. 
FAILURE RATE, ASSESSED. The failure rate of an item determined as a limiting value or values of the confidence interval with a stated confidence level, based on the same data as the observed failure rate of nominally identical items.
FAILURE RATE, EXTRAPOLATED. Extension by a defined extrapolation or interpolation of the observed or assessed failure rate for durations and/or conditions different from those applying to the conditions of that observed or assessed failure rate.
FAILURE RATE, OBSERVED. The ratio of the total number of failures in a sample to the total cumulation observed time on that sample. The observed failure rate is to be associated with particular, and stated time intervals (or summation of intervals) in the life of the items, and with stated conditions.
FAILURE RATE, PREDICTED. For the stated conditions of use and the design considerations of an item, the failure rate computed from the observed, assessed or extrapolated failure rates of its parts. 
FAILURE, SECONDARY. Failure of an item caused either directly or indirectly by the failure of another item. 
FAILURE, SUDDEN. Failures that could not be anticipated by prior examination. 
FAILURE, WEAR-OUT. A failure which occurs as a result of deterioration processes or mechanical wear and whose probability of occurrence increases with time. 
FAIR GAME. In probability theory, a game consisting of a sequence of trials is deemed to be a “fair” game if the cost of each trial is equal to the expected value of the gain from each trial. In game theory, a game which with proper play neither adversary has an advantage.
FARKAS' LEMMA. If for every solution of WA ≤ 0 it is also true that Wb ≤ 0, then there exists a vector X ≥ 0 such that Ax = b; and thus (WA)X = Wb. (If a linear homogeneous inequality Wb ≤ 0 holds for all W satisfying a system of homogeneous inequalities WA ≤ 0, then the inequality can be expressed as a non-negative combination of the inequalities of the system WA ≤ 0.) 
FATHOM. A term used in branch and bound algorithms to indicate a node has been fully explored; i.e., it has been determined that the node cannot contain a solution better than the incumbent.
FEASIBLE BASIS. A basis yielding a basic feasible solution (q.v). [ 19]
FEASIBLE SOLUTION. A solution satisfying the constraints of A mathematical programming problem in which all variables are non-negative. 
FINITE AND INFINITE GAMES. A game is finite if each player has only a finite number of possible pure strategies; it is infinite if at least one player has an infinite number of possible pure strategies (e.g., a pure strategy might ideally consist of choosing an instant from a given interval of time at which to fire a gun). [2 1]
FRACTIONAL PROGRAMMING. A class of mathematical programming problems in which the objective function is the quotient of linear functions. 
FRAME. The list of units, or items, accessible for test. Each unit has a serial number associated with it, actually or conceptually. If there are units in the population that are not covered by the frame, statistical inferences (estimates, confidence limits, etc.) refer to the frame, not the population. Generalizations from the frame to the population must be based on judgment. 
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