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Given exponential a random distribution X with PDF $f_X(x)=\lambda e^{-\lambda x}$ and a random variable Z with the PMF $p_Z[z]=0.5, z= \pm1$, I am trying to find the PDF of $Y=ZX$ (I also know that Z and X are independent).

The fact that Z is discrete is throwing me off a bit. My guess is that $$f_{Y}(y)=f_x(y)P[Z=1]+f_x(y)P[Z=-1]=\lambda e^{-\lambda y}$$

or is it just: $f_{Y}(y)=f_X(y)p_Z[y]$?

Thanks!

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  • $\begingroup$ There is no joint pdf, contrary to the suggestion in your title. $\endgroup$
    – whuber ♦
    Commented Dec 16, 2020 at 21:59
  • $\begingroup$ Are you suggesting that i find the joint pdf of $f_{X,Y}(x, y)$ and then calculate marginal PDF $f_Y(y)$? $\endgroup$
    – darisoy
    Commented Dec 16, 2020 at 22:09
  • $\begingroup$ I am saying a joint pdf doesn't exist, but your title implies it does. $\endgroup$
    – whuber ♦
    Commented Dec 16, 2020 at 22:11
  • $\begingroup$ You could change your title to ask for the joint distribution. There are many similar questions, see through this list $\endgroup$
    – kjetil b halvorsen ♦
    Commented Dec 18, 2020 at 15:33

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