Probabilistic Analysis
Given any random variables X1, X2..., Xn, the order statistics X(1), X(2), ..., X(n) are also random variables, defined by sorting the values (realizations) of X1, ..., Xn in increasing order.
When the random variables X1, X2..., Xn form a sample they are independent and identically distributed. This is the case treated below. In general, the random variables X1, ..., Xn can arise by sampling from more than one population. Then they are independent, but not necessarily identically distributed, and their joint probability distribution is given by the Bapat–Beg theorem.
From now on, we will assume that the random variables under consideration are continuous and, where convenient, we will also assume that they have a probability density function (that is, they are absolutely continuous). The peculiarities of the analysis of distributions assigning mass to points (in particular, discrete distributions) are discussed at the end.
Read more about this topic: Order Statistic
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“Ask anyone committed to Marxist analysis how many angels on the head of a pin, and you will be asked in return to never mind the angels, tell me who controls the production of pins.”
—Joan Didion (b. 1934)