ABSTRACT
Let X1,X2, ... be a sequence of independent r.v.’s with respective d.f.’s F1(x),F2(x), ... , and let
W˜n = max{X1, ...,Xn} and W ˜
n = min{X1, ...,Xn}. The distributions of W˜n and W
˜ n have simple representations in terms of the d.f.’s Fi. Namely,
since Xi’s are independent,
P(W˜n ≤ x) = P(X1 ≤ x, ...,Xn ≤ x) = n∏
i=1 P(Xi ≤ x) =
Fi(x), (1.1.1)
and
P(W ˜
n > x) = P(X1 > x, ...,Xn > x) = n∏
i=1 P(Xi > x) =
(1−Fi(x)). (1.1.2)
For the d.f. of the minimum, we have
P(W ˜
n ≤ x) = 1−P(W ˜
n > x) = 1− n∏
i=1 (1−Fi(x)). (1.1.3)
If the r.v.’s Xi are identically distributed and Fi(x) = F(x), then (1.1.1) implies
P(W˜n ≤ x) = Fn(x), (1.1.4)
and from (1.1.3), it follows that
P(W ˜
n ≤ x) = 1− (1−F(x))n. (1.1.5)
EXAMPLE 1 (The last survivor annuity). The term annuity means a sequence of payments. An elderly couple begins to get regular retirement annuity which terminates only when both spouses die. Suppose that X1,X2, the remaining lifetimes of the husband and
(a) (b) FIGURE 1.
