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Let's assume that I have a Standard Normal variable $X\sim\mathcal N(0,1)$. I am looking for a variable $Y$ so that $Z=XY$ follows a normal distribution too (regardless of its mean and variance).

Which distribution should $Y$ follow in order to fulfil this requirement?

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    $\begingroup$ $Y\equiv c, c\not= 0$ $\endgroup$ Commented May 12, 2017 at 8:42
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    $\begingroup$ Ok, that's fine. But I would appreciate anyway something more random :). $\endgroup$ Commented May 12, 2017 at 8:51
  • $\begingroup$ Maybe you should reformulate your question and ask: "... if $XY$ has normal distribution then what can be concluded about the distribution of $Y$?" $\endgroup$ Commented May 12, 2017 at 8:58

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One step above the trivial $Y\equiv c$: you can take $Y=c$ with probability $p$ and $Y=-c$ with probability $(1-p)$ (for any $c\neq 0$ and $p\in(0,1)$).

(As Gono points out, this assumes $X$ and $Y$ are independent.)

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  • $\begingroup$ That's not true … take $Y = sgn(X)$ then $$P(Y = 1) = P(X > 0) = 0.5 = p$$ and $$P(Y = -1) = P(X < 0) = 0.5 = 1-p$$ but $Z = XY = |X|$ is not normally distributed… $\endgroup$ Commented May 12, 2017 at 10:51
  • $\begingroup$ Yes, true, I was assuming $Y$ had to be independent of $X$. If not, it is easy: just take $Y=Z/X$ where $Z$ is any normal variable you like ($Z$ and $X$ don't need to be independent). $\endgroup$ Commented May 12, 2017 at 10:55

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