The random variables $X_1,X_2,...X_n,Y_1,Y_2,...Y_n $ are independent and $U(0,a)$-distributed. Determine the distribution of
$$Z_n=n\cdot \log\left(\frac{\max\left(X_{(n)},Y_{(n)}\right)}{\min\left(X_{(n)},Y_{(n)}\right)}\right)$$
where $X_{(n)},Y_{(n)}$ denotes $n$th order variable.
This problem is giving me a headache. The answer is that $Z_n$ is $\text{Exp}(1)$-distributed. But i can't derive that conclusion. I'd like to show you my derivation, but its so long and contains so many integrals and fractions that it would take a while to write it here. But basically my approach was to derive the distribution of $M=\max(X_{(n)},Y_{(n)})$ and $L=\min(X_{(n)},Y_{(n)})$. I get the following cdf's:
$F_M=(F_X\cdot F_Y)^n=(F_X)^{2n}$ and $F_L=1-(1-(F_X)^n)\cdot (1-(F_Y)^n)=1-(1-(F_X)^n)^2$ ,
correct? $F_X $ stands for the cdf of $X_i$
Then i define the random variable $T=\frac{M}{L}$ and compute its cdf to finally be able to derive the distribution of $Z_n=n\cdot \log(T)$. But thats where i think everything goes wrong.
Any help appreciated, thanks.
To go a step further from the comment,
$$\log \frac{X_{(n)}}{Y_{(n)}} = \log X_{(n)} - \log Y_{(n)},$$ hence we need only find the distribution of the maximum order statistic. Assuming without loss of generality that $a = 1$ hence $$F_{X_{(n)}}(x) = \prod_{i=1}^n \Pr[X_i \le x] = F_X(x)^n = x^n,$$ and it follows that $$f_{X_{(n)}}(x) = nx^{n-1}, \quad 0 \le x \le 1.$$ Note $Y_{(n)}$ also follows the same distribution. Continuing, $W = \log X_{(n)}$ has density $$f_W(w) = f_{X_{(n)}} (e^w) e^w = ne^{w(n-1)} e^w = ne^{nw}, \quad -\infty < w \le 0;$$ that is to say, $-W$ is exponential with rate $n$. The rest is quite straightforward and is left as an exercise.