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r19ecua
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Homework Statement
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
Homework Equations
[itex]\Phi[/itex]X([itex]\omega[/itex]) = E[ejωx] = ∫fX(x)ejωxdx
[itex]\Phi[/itex]Z(ω) = [itex]\Phi[/itex]X([itex]\omega[/itex]) * [itex]\Phi[/itex]Y([itex]\omega[/itex])
fZ(z) = 1/ (2[itex]\pi[/itex]) ∫ [itex]\Phi[/itex]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..
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