Say I have two normal independent random variables with nonzero means - X distributed as N(a,b) and Y as N(c,d). What is the mean of the product XY?

I see that the distribution has been computed (Is the product of two Gaussian random variables also a Gaussian?) as a bessel function in an earlier question if it is 0 mean- but if the mean is not zero, is there a simple expression for E[XY] ?

  • $\begingroup$ Then the result is trivial: $E[XY]=E[X]E[Y]$ under independence of $X$ and $Y$ $\endgroup$ – sinbadh Mar 4 '16 at 21:53
  • $\begingroup$ I see - if it is not independent, and the correlation is some value say 'r', is there a simple expression? $\endgroup$ – user1775614 Mar 4 '16 at 21:55
  • $\begingroup$ Yes. Remember $Cov(X,Y)=E[XY]-E[X]E[Y]$. $\endgroup$ – sinbadh Mar 4 '16 at 21:57
  • $\begingroup$ got it. So it would just be the product + r $\endgroup$ – user1775614 Mar 4 '16 at 22:02
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    $\begingroup$ Almost. The general formula is $E[XY]=E[X]E[Y] + r\sigma_X\sigma_Y$ [writing $\sigma_X:=\sqrt{Var(X)}$] $\endgroup$ – grand_chat Mar 4 '16 at 22:16

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