I know that sum of 2 correlated normal random variables are written as $X+Y\sim N(\mu_x+\mu_y,\sigma^2_x+\sigma^2_y+2\rho_{xy}\sigma_x\sigma_y)$, I am now wondering how is 3 correlated normal random variables are constructed. Is it in the form of $X+Y+Z\sim N(\mu_x+\mu_y+\mu_z, \sigma^2_x+\sigma^2_y+\sigma^2_z+2\rho_{xy}\sigma_x\sigma_y+2\rho_{yz}\sigma_y\sigma_z+2\rho_{xz}\sigma_x\sigma_z)$? Thank you so much!

  • $\begingroup$ You've got the right tools infront of you. Define $W = X+Y$, and you know that $W \sim N(\mu_x + \mu_y, ...)$. Define $\mu_{x+y} = \mu_x + \mu_y$, and likewise for the variance. Now apply your same formula to $W + Z$. Does that make sense? $\endgroup$ Oct 31, 2020 at 16:21
  • $\begingroup$ @Cam.Davidson.Pilon Thanks for the hint! I can expand everything fine but kind of being stuck at $\rho_{wz}$ $\endgroup$
    – gegege
    Oct 31, 2020 at 16:39
  • $\begingroup$ Ah, you may have to expand the correlation coefficient into it's covariance form: $\rho_{x+y, z} = \frac{Cov(X+Y, Z)}{\sigma_{X+Y} \sigma_{Z}}$ and try to simplify from there (i..e simplify the term $\rho_{x+y, z}\sigma_{X+Y} \sigma_{Z}$) $\endgroup$ Oct 31, 2020 at 17:20
  • $\begingroup$ Your guess is correct (see variance of a sum of random variables), since $\text{Cov}(X,Y) = \rho_{xy} \sigma_x \sigma_y$. $\endgroup$
    – angryavian
    Oct 31, 2020 at 17:53
  • 1
    $\begingroup$ Use a vectorial approach:$$X+Y+Z=(X\ Y\ Z)\cdot(1\ 1\ 1)^\text{T}$$ $\endgroup$
    – Xi'an
    Nov 1, 2020 at 12:31


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