Let the random variables $X_1,\,X_2,\,\ldots,\,X_K$ be i.i.d. exponential random variables with parameter 1. Also, let the random variables $Y_1,\,Y_2,\,\ldots,\,Y_K$ be defined similarly.
Now let
$$Y_{(1)}\leq Y_{(2)}\leq\cdots \leq Y_{(K)}$$
I want to find the following CDF
$$F_Z(z)=\text{Pr}\left[\max\left(\frac{X_{i_{K-1}}}{Y_{(K-1)}},\,\frac{X_{i_{K}}}{Y_{(K)}}\right)\leq z\right]$$
where $i_k$ is the index corresponding to the $k$th order statistic.
My attempt:
$$F_Z(z)=\int_{x_1=0}^{\infty}\int_{x_2=0}^{\infty}\int_{u=0}^{x_1/z}\int_{v=u}^{\max(u,\,x_2/z)}f_{(K-1),(K)}(u,\,v)f_X(x_1)f_X(x_2)\,dudvdx_1dx_2$$
where $f_{(K-1),(K)}(u,\,v)$ is the joint PDF of the RVs $Y_{(K-1)}$ and $Y_{(K)}$, and $f_X(x_k)$ is the PDF of the random variable $X_k$.
My question is: did I get the limits right?