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

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


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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.