Hello everyone,

I have faced a difficult function of two random variables. The problem is described as follows.

Given two independent exponential random variables $\displaystyle x$ and $\displaystyle y$, $\displaystyle x \ge 0$ and $\displaystyle y \ge 0$, with the corresponding PDF functions as $\displaystyle {f_x}(x)$ and $\displaystyle {f_y}(y)$. The random variable $\displaystyle u = \min \{ \frac{I}{x},P\} \times y$, where $\displaystyle I$ and $\displaystyle P$ are constants, has the CDF function written as:

$\displaystyle \Pr \{ u \le U\} = \int\limits_{x = I/P}^\infty \int\limits_{y = 0}^{x.U/I} {{f_x}(x){f_y}(y)dydx + } \int\limits_{x = 0}^{I/P} {} \int\limits_{y = 0}^{U/P} {{f_x}(x){f_y}(y)dydx} $

I cannot understand how the CDF of $\displaystyle u$ is obtained as above. Can everyone give me some tips for solving the problem?. Thanks very much.