This is a question from a textbook of nonparametric statistics by gibbons.
A manufacturer wants to market a new brand of heat-resistant tiles which may be used on the space shuttle. A random sample of size $m$ of these tiles is put to the test and the heat resistance capacities of the tiles are measured. Let $X(1)$ denote the smallest of these measurements. The manufacturer is interested in finding the probability that in a future test (performed by, say, an independent agency) of a random sample of $n$ of these tiles at least $k$ ($k = 1, 2, . . . , n$) will have a heat resistance capacity exceeding $X(1)$ units. Assume that the heat resistance capacities of these tiles follows a continuous distribution with CDF $F$.
the question is:
Show that this probability is given by $\sum^n_{r=k}P(r)$
where $P(r)=\frac{mn!(r+m-1)!}{r!(n+m)!}$
I don't now where to start with this question. I know that the CDF F has a binomial distribution. But I don't see how that leads to this answer. Can somebody put me in the right direction?
Has this something to do with p-th quantile of order statistics?
Suppose we order all the heat resistance capacities of the $m+n$ tiles of both samples from left to right, lowest to highest. Write the capacities of the $m$ tiles of the first sample in blue and the capacities of the second sample of $n$ tiles in red. Since the capacities follow a continuous distribution, we may assume there are no ties. If we ignore the values of the numbers and simply base our analysis on whether each number is red or blue, there are $\binom{m+n}{m}$ possible arrangements of colors, each of which is equally likely.
We would like to count the number of arrangements in which exactly $r$ red numbers (which correspond to the second sample) follow the first blue number. From left to right, there must be $n-r$ red numbers, then the first blue number, then $r+m-1$ numbers of which $r$ are red. So, again ignoring the values of the numbers and looking only at their colors, there are $\binom{r+m-1}{r}$ possible arrangements.
So the probability that exactly $r$ red numbers follow the first blue number, i.e. exactly $r$ tiles have heat capacities exceeding $X_{(1)}$, is $$P(r) = \frac{\binom{r+m-1}{r}}{\binom{m+n}{m}} = \frac{m \;n! (r+m-1)!}{r! (m+n)!}$$