Order statistics when variables have different distributions

Question

Let a,b and c be random variables, where a~U[0,8] and b~U[3,8]. Let c=max{a,b}, What is the mean of c?

In general, let $(x_1, ..., x_n)$ be independently distributed in different supports. What is the distribution of its maximum, i.e. of its highest order statistic?

I have the solution for my first question (from simulating it in Matlab) but I am looking for the general expression. I encountered the Bapat-Beg Theorem but I am convinced there is something simpler I can use. Intuitive explanations are very welcomed.

Answer 1

First, it is clear that the support of $c$ must be $(3,8).$ The maximum must take values in this interval.

Second, the cdf of $c$ must be $$F_c(t) = P(c \le t) = P(a \le t, b \le t) = P(a \le t)\;P(b \le t) = \cdots ,$$ for $3 < t < 8.$ This is a very common way to find the distribution of the maximum of two independent random variables.

Third, find the density function and integrate to find the mean.

This is a case in which simulation gives you a pretty good idea of the answer as a check on your analysis.

 m = 10^6;  a = runif(m, 0, 8);  b = runif(m, 3, 8)
 c = pmax(a,b);  hist(c, prob=T);  mean(c)
 ## 6.022239

The histogram shows a trapezoidal PDF, and the simulation gives a reasonably good estimate of the expectation (maybe one or two place accuracy).