How Do You Find the PDF of a Ratio of Exponential Random Variables?

[dionysian]

Homework Statement

Let X and Y be two independent random variables each exponentially distributed with parameter 1. Define a new random variable:

$$z = \frac{x}{{x + y}}$$

Find the PDF of Z

Homework Equations

The Attempt at a Solution

$$\begin{array}{l}<br /> {F_Z}(z) = P(Z < z) = P\left( {\frac{x}{{x + y}} < z} \right) = P\left( {x \le \frac{{zy}}{{1 - z}}} \right) \\ <br /> {F_Z}(z) = \int\limits_0^\infty {\int_0^{\frac{{zy}}{{1 - z}}} {{f_{xy}}(x,y)dxdy} } \\ <br /> {f_{xy}}(x,y) = {f_x}(x){f_y}(y) \\ <br /> {F_Z}(z) = \int\limits_0^\infty {\int_0^{\frac{{zy}}{{1 - z}}} {{f_x}(x){f_y}(y)dxdy = } } \int\limits_0^\infty {\int_0^{\frac{{zy}}{{1 - z}}} {{e^{ - x}}{e^{ - y}}dxdy} } = \int\limits_0^\infty {{e^{ - y}}\left[ {\int_0^{\frac{{zy}}{{1 - z}}} {{e^{ - x}}dx} } \right]} dy \\ <br /> {F_Z}(z) = \int\limits_0^\infty {{e^{ - y}}\left[ { - {e^{ - \frac{{zy}}{{1 - z}}}} + 1} \right]} dy = \int\limits_0^\infty { - {e^{ - y}}{e^{ - \frac{{zy}}{{1 - z}}}} + {e^{ - y}}} dy = \int\limits_0^\infty { - {e^{ - \frac{{y(1 - z) - zy}}{{1 - z}}}} + {e^{ - y}}} dy \\ <br /> {F_Z}(z) = \int\limits_0^\infty { - {e^{ - \frac{y}{{1 - z}}}} + {e^{ - y}}} dy = (1 - z){e^{ - \frac{y}{{1 - z}}}}|_0^\infty - {e^{ - y}}|_0^\infty = z \\ <br /> \end{array}$$
Now i know that if i take the derivative of this i will get the "pdf" but its obviously wrong. Any thoughts?

 

[jbunniii]

$$P\left( {\frac{x}{{x + y}} < z} \right) = P\left( {x \le \frac{{zy}}{{1 - z}}} \right)$$

This step is invalid if $z > 1$. (The inequality gets reversed in that case.)

But you could instead write

$$P\left( \frac{x}{x+y} < z \right) = P\left(y > \frac{x(1-z)}{z}\right) = 1 - P\left(y \leq \frac{x(1-z)}{z}\right)$$

I'm not sure if that will be any more helpful, but at least it's correct.

I wonder if it would be helpful to work with the reciprocal:

$$\frac{1}{z} = \frac{x + y}{x} = 1 + \frac{y}{x}$$

It shouldn't be hard to work out the pdf of

$$\frac{y}{x}$$

as it is the quotient of two independent random variables. Adding 1 just shifts the pdf to the right by 1. Then do you know how to find the pdf of the reciprocal of a random variable with known pdf?