If $X, Y$ and $Z$ are non identical and independent exponential random variables, what is the probability density function of $X + Y - Z$? [closed]

Nothing to add at this point, I get what $X + Y$ PDF is but I don't know how to proceed

If $X,\,Y$ have different means, the characteristic function is$$\frac{1}{(1-i\mu_Xt)(1-i\mu_Yt)(1+i\mu_Zt)}=\frac{A_X}{1-i\mu_Xt}+\frac{A_Y}{1-i\mu_Yt}+\frac{B_Z}{1+i\mu_Zt}$$(coefficients are an exercise), so the PDF of $W:=X+Y-Z$ is$$\frac{A_X}{\mu_X}\exp(-w/\mu_X)[w\ge0]+\frac{A_Y}{\mu_Y}\exp(-w/\mu_Y)[w\ge0]-\frac{B_Z}{\mu_Z}\exp(w/\mu_Z)[w\le0].$$Feel free to instead parameterize the exponential distributions as $\lambda_X=\mu_X^{-1}$ etc. As for the case where $X,\,Y$ have the same mean, you can treat that in a similar way, using the fact that $\frac{1}{(1-\theta t)^2}$ is the characteristic function of a distribution with PDF $\frac{u}{\theta^2}\exp(-u/\theta)$ on $[0,\,\infty)$.