A complicated function of two random variables

**lptuyen** · May 30th 2011, 06:33 PM

Hello everyone,

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

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

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

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

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