We introduce variates (random variables), defining their cumulative distribution function (c.d.f.) and probability density function (p.d.f.). We give the two main decomposition theorems of c.d.f.s, Jordan and Lebesgue, and the unifying Stieltjes integral approach for continuous and discrete variates. We study properties and descriptions such as symmetry, median, quantiles, and the quantile function, and modes of a distribution. The mixing of variates is also studied, including the famous example of Student's t, which can be represented as a mixed-normal variate.
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