Did I correctly find the probability density function?

[r19ecua]

Homework Statement

A random variable x has a probability density function given by

f_X(x) = e^-x , x ≥ 0

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

f_Y(y) = e^y , y ≤ 0

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

Homework Equations

\Phi_X(\omega) = E[e^jωx] = ∫f_X(x)e^jωxdx

\Phi_Z(ω) = \Phi_X(\omega) * \Phi_Y(\omega)

f_Z(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

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..

 

[Ray Vickson]

Homework Statement

A random variable x has a probability density function given by

f_X(x) = e^-x , x ≥ 0

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

f_Y(y) = e^y , y ≤ 0

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

Homework Equations

\Phi_X(\omega) = E[e^jωx] = ∫f_X(x)e^jωxdx

\Phi_Z(ω) = \Phi_X(\omega) * \Phi_Y(\omega)

f_Z(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

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..

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

 

[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:

e^z / 2 .. .. Is that correct?

 

[Ray Vickson]

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:

e^z / 2 .. .. Is that correct?

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