Continuous univariate distributions

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Continuous univariate distributions

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Continuous univariate statistical distributions are functions that describe the probability that a random variable, X say, lies in a given range. For example, let the probability that X lies in the range [a,b] be given by P(aX≤b). However, if X can take any specific value on the real line, the probability of any specific value is effectively zero (since we would have a=b, so no range). For this reason it is common to describe continuous probability distributions in terms of their cumulative distribution function, F(x), and then to define the distribution function (or probability density function) by the differential f(x)=dF(x)/dx. Adopting this approach, we have:

where t is a dummy variable introduced for convenience. We may also write:

thus P(aX≤b)=F(b)-F(a).

The following are examples of well-known continuous probability distributions, each of which is described in further detail in the subsections below: Beta, chi-square, Erlang, Exponential, F, Gamma, Normal, Pareto, Student's t, Uniform, von Mises, and Weibull.