X=Normal(0,1) random var. Y=Uniform{-1,1} random var. Show that Z=X.Y is normal random variable.

Thanks for your help in advance.

I tried to solve it through double integral but it failed to do. There should be a trick to solve this.

  • 2
    $\begingroup$ The conclusion is false. Did you perhaps mean to stipulate that $Y$ is a random variable that takes the values $\{-1,1\}$ with equal probability (of $1/2$ each)? Regardless, please consult our help center about asking questions and kindly tell us what attempts you have made at this question and where specifically you could use some help. $\endgroup$ – whuber Feb 5 '14 at 23:06
  • $\begingroup$ Yes this is what I mean by U(1,-1).Sorry for inconvenience but isn't it U(-1,1) general notation for that? $\endgroup$ – arke Feb 6 '14 at 0:04
  • $\begingroup$ @arke No, it is $U\{-1, +1\}$ $\endgroup$ – Dilip Sarwate Feb 6 '14 at 0:19
  • $\begingroup$ arke, "$U(-1,1)$" without further adornment is generally taken to be a continuous uniform distribution between $-1$ and $1$ ... for which the statement is untrue. If you want a 50-50 chance of the values -1 and 1, you'd specify it differently. $\endgroup$ – Glen_b -Reinstate Monica Feb 6 '14 at 0:41
  • $\begingroup$ Thanks all helping me correct my notation.However, do have any idea why the conclusion is wrong if it is U(-1,1)(I know now it is uniform but I just wonder there is any proof also for that." $\endgroup$ – arke Feb 6 '14 at 0:56

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