# Calculate the characteristic function $\varphi_W$ of W

$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise.

I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.

$$\phi_W(t)=\langle e^{\mathrm{i}tx}\rangle=\int_0^\infty dx\ x e^{-x}e^{\mathrm{i}tx}=-\mathrm{i}\frac{d}{dt}\int_0^\infty dx\ e^{-(1-\mathrm{i}t)x}=-\mathrm{i}\frac{d}{dt}\frac{1}{1-\mathrm{i}t}=\frac{1}{(1-\mathrm{i}t)^2}\ .$$
• Let $Z=X+Y$. Then $$p(z)=\int_0^\infty dx\int_0^\infty dy e^{-x-y}\delta(z-(x+y))=\int_0^z dx\ e^{-x}e^{-(z-x)}=z e^{-z}\ ,\mathrm{for}\quad z\geq 0\ .$$ – Pierpaolo Vivo Feb 9 '16 at 13:31