# Did I correctly find the probability density function?

1. Mar 28, 2013

### r19ecua

1. The problem statement, all variables and given/known data
A random variable x has a probability density function given by

fX(x) = e-x , x ≥ 0

and an independent random variable Y has a probability density function of

fY(y) = ey , y ≤ 0

using the characterisic functions, find the probability density function of Z = X + Y

2. Relevant equations

$\Phi$X($\omega$) = E[ejωx] = ∫fX(x)ejωxdx

$\Phi$Z(ω) = $\Phi$X($\omega$) * $\Phi$Y($\omega$)

fZ(z) = 1/ (2$\pi$) ∫ $\Phi$Z(ω) * e-jωz

the integrals within section 2 are from negative infinity to positive infinity . . .
also it's 1 over 2 pi . . didn't know how to make that into a fraction

3. The attempt at a solution

My attempt was scanned and uploaded via photobucket.. here's the scan!
http://i359.photobucket.com/albums/oo39/r19ecua/scan0356.jpg

I'm wondering if Cauchy's was a good way to handle this..

Last edited: Mar 28, 2013
2. Mar 28, 2013

### Ray Vickson

Since $Z$ is a real-valued random variable its pdf $f_Z(z)$ cannot be complex for real $z .$ So your solution is incorrect.

3. Mar 28, 2013

### r19ecua

I just realized that I did not replace the 1's with j's which led to my grand error! The j's are meant to cancel.. my answer should be:

ez / 2 .. .. Is that correct?

4. Mar 29, 2013

### Ray Vickson

No, obviously not. Does e^z have a finite integral when you integrate z from -∞ to +∞?