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

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

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1
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$$ \phi_W(t)=\langle e^{\mathrm{i}tx}\rangle=\int_0^\infty dx\ x e^{-x}e^{\mathrm{i}tx}=-\mathrm{i}\frac{d}{dt}\int_0^\infty dx\ e^{-(1-\mathrm{i}t)x}=-\mathrm{i}\frac{d}{dt}\frac{1}{1-\mathrm{i}t}=\frac{1}{(1-\mathrm{i}t)^2}\ . $$

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  • $\begingroup$ Thank you and how do you find pdf of X+Y, where X and Y are independent exponential r.v.s, each with parameter 1. $\endgroup$ – user164945 Feb 9 '16 at 13:17
  • $\begingroup$ @user164945, please post another question. And if my answer was what you were looking for, don't forget to accept it. Thanks ;-) $\endgroup$ – Pierpaolo Vivo Feb 9 '16 at 13:19
  • $\begingroup$ Forgot about that :P Hmm it's related question tho, if you could edit and answer it, I would be really appreciated. $\endgroup$ – user164945 Feb 9 '16 at 13:24
  • $\begingroup$ You haven't accepted, just upvoted ;-) Will see what I can do... $\endgroup$ – Pierpaolo Vivo Feb 9 '16 at 13:25
  • 1
    $\begingroup$ Let $Z=X+Y$. Then $$ p(z)=\int_0^\infty dx\int_0^\infty dy e^{-x-y}\delta(z-(x+y))=\int_0^z dx\ e^{-x}e^{-(z-x)}=z e^{-z}\ ,\mathrm{for}\quad z\geq 0\ . $$ $\endgroup$ – Pierpaolo Vivo Feb 9 '16 at 13:31

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