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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]
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.
Independent: Each outcome will not affect the other outcome (for from 1 to 10), which means the variables , …, are independent of each other. Identically distributed : Regardless of whether the coin is fair (with a probability of 1/2 for heads) or biased, as long as the same coin is used for each flip, the probability of getting heads remains ...
Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables X and Y are independent if and only if the random vector ( X , Y ) with joint cumulative distribution function (CDF) F X , Y ( x , y ) {\displaystyle F_{X,Y}(x,y)} satisfies
The marginal probability is the probability of a single event occurring, independent of other events. A conditional probability, on the other hand, is the probability that an event occurs given that another specific event has already occurred. This means that the calculation for one variable is dependent on another variable. [2]
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 ...
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.
In probability theory and theoretical computer science, McDiarmid's inequality (named after Colin McDiarmid [1]) is a concentration inequality which bounds the deviation between the sampled value and the expected value of certain functions when they are evaluated on independent random variables.
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