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In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) [5] and providing an output (which may also be a number). [5] A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. [6]
The argument of a hyperbolic function is a hyperbolic angle. A mathematical function has one or more arguments in the form of independent variables designated in the definition, which can also contain parameters. The independent variables are mentioned in the list of arguments that the function takes, whereas the parameters are not.
Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent. [23] The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic.
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
t may contain some, all or none of the x 1, …, x n and it may contain other variables. In this case we say that function definition binds the variables x 1, …, x n. In this manner, function definition expressions of the kind shown above can be thought of as the variable binding operator, analogous to the lambda expressions of lambda calculus.
The characteristic function approach is particularly useful in analysis of linear combinations of independent random variables: a classical proof of the Central Limit Theorem uses characteristic functions and Lévy's continuity theorem. Another important application is to the theory of the decomposability of random variables.
In that model, the random variables X 1, ..., X n are not independent, but they are conditionally independent given the value of p. In particular, if a large number of the X s are observed to be equal to 1, that would imply a high conditional probability , given that observation, that p is near 1, and thus a high conditional probability , given ...
An infinite set of vectors is linearly independent if every nonempty finite subset is linearly independent. Conversely, an infinite set of vectors is linearly dependent if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.