Probability of two random uniform variables

#1
Suppose X is uniform [e,f] and Y is uniform [g,h]. We want to find the pdf of the X+Y for the general uniform random variables.

Using the characteristic equation we multiply both together and we need to take the inverse Fourier transformation. I do not know how to simplify it. Is there a trick I am not seeing?
 

Dason

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#2
That seems like an awfully complicated method to use to derive this distribution. Is there a particular reason you're using that approach?
 
#3
Yes, in this case, the characteristic function approach will only take extra steps to prove the necessity of taking an integral. So you may as well do it from the beginning: the cdf of X+Y equals

F(z) = Prob(X+Y <= z) = Int_e^f f_X(x) * Prob(X+Y <= z | X = x) dx = Int_e^f 1/(f-e) * Prob(Y <= max(z-x,0) | X = x) dx = ...