Often, we like to arrange numbers in a decreasing order for ease of discussion (and mathematical analysis). When there are two values V1 and V2, we denote by V1:2 the higher of the two values, and by V2:2 the lower of the two values. In general, if there are n values we denote the highest, the second-highest, etc., by V1:n,V2:n,…,Vn:n. For instance, if V1=8 and V2=10 then we have V1:2=10 and V2:2=8. Similarly, we can also arrange bids and denote them similarly.
V1:n is called the highest or first order statistic, V2:n is called the second-highest or simply the second order statistic, and so on.
Order Statistics of Random Variables
We tend to be interested in order statistics of a given number of random variables. For example, think of the 40 students who will take my class next semester. Suppose I wish to estimate the height of the tallest student. The heights of the students would be denote by X1, X2, …,X40 and the height of the tallest student max{X1, X2, …,X40} would be the highest order statistic X1:40 for these random variables. So, in what I am trying to do, I must calculate the expected value of E[X1:40]. Similarly, I may want to calculate the expected height of the second tallest student given by E[X2:40] and so on.
In what we do, we mostly consider random variables that are independently and identically distributed according to some distribution function F(.). The calculation of E[X1:40] requires the knowledge of the probability distribution function F1:40(.) of X1:40. It is easy to calculate this distribution function. You simply observe that the random variables are independent and identically distribution so we have the following:
F1:40(x) = P[X1:40≤x] = P[X1≤x & X2 ≤ x & …& X40≤x] = P[X1≤x] P[X2 ≤ x] … P[X40≤x] = F(x)40
If the individual heights are random variables that are uniformly distributed between 0 and 1 (ridiculously unlikely, but this is only an illustration!) then F(x)=x so that F1:40(x)=x40.
You can similarly calculate F2:40(x) and so on and then use them to calculate the expected values the usual way.