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In statistics, the mode is the value that appears most often in a set of data values. [1] If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e., x=argmax x i P(X = x i)). In other words, it is the value that is most likely to be sampled.
Usually this definition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any single value is zero, while this definition allows for a non-zero probability, or an "atom of probability", at the mode.
Mode: for a discrete random variable, the value with highest probability; for an absolutely continuous random variable, a location at which the probability density function has a local peak. Quantile : the q-quantile is the value x {\displaystyle x} such that P ( X < x ) = q {\displaystyle P(X<x)=q} .
Moreover, the negative of the mode is exactly the mode for a noncentral t-distribution with the same number of degrees of freedom ν but noncentrality parameter −μ. The mode is strictly increasing with μ (it always moves in the same direction as μ is adjusted in). In the limit, when μ → 0, the mode is approximated by
A chart showing a uniform distribution. In probability theory and statistics, a collection of random variables is independent and identically distributed (i.i.d., iid, or IID) if each random variable has the same probability distribution as the others and all are mutually independent. [1]
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is ...
When the image (or range) of is finitely or infinitely countable, the random variable is called a discrete random variable [5]: 399 and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of .
For a discrete probability distribution with a finite number n of indices at which the probability mass function f takes non-zero values, the basic sampling algorithm is straightforward. The interval [0, 1) is divided in n intervals [0, f(1)), [f(1), f(1) + f(2)), ... The width of interval i equals the probability f(i).