Supposing $X$ and $Y$ are random variables with a joint bivariate normal distribution and covariance matrix $\Sigma_{XY}$. Consider the following linear combination for constants $A$, $B$ and $C$:

$$Z = AX + BY + C$$

This Cross Validated post states that $Y$ will be normally distributed according to:

$$ AX+BY+C \sim \mathcal{N}\left[ \left(\begin{matrix}A& B \end{matrix}\right) \left(\begin{matrix}\mu_X\\\mu_Y\end{matrix}\right) + C, \left(\begin{matrix}A & B \end{matrix}\right)\Sigma_{X,Y} \left(\begin{matrix}A^T \\ B^T \end{matrix}\right)\right] $$

However I would like to know how to compute the covariance matrix $\Sigma_{X,Y,Z}$. How can I calculate the correlation between the new variable $Z$ and the old variables $X$ and $Y$ i.e. $corr(Z,X)$ and $corr(Z,Y)$?

  • $\begingroup$ For those formulas to be true, should be "...with covariance matrix $\Sigma_{XY}$, not correlation matrix. $\endgroup$ – Matthew Gunn Jun 3 '16 at 16:42
  • $\begingroup$ Correct. I have edited it. $\endgroup$ – 7Jack Jun 3 '16 at 17:00
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    $\begingroup$ @MatthewGunn Even with the change you have suggested, the formulas are correct only as far as the mean and variance are concerned, but the $\mathcal N (\cdot, \cdot)$ part is not correct unless one makes the stronger assumption that $X$ and $Y$ are jointly normal. See whuber's comment on Xi'an's answer in the thread referenced by the OP.. $\endgroup$ – Dilip Sarwate Jun 3 '16 at 19:49
  • $\begingroup$ @DilipSarwate I did not appreciate that. This post explains it fairly well. So only linear combinations of independent normal variables are guaranteed to be normal. If they are correlated this is no longer the case. So I will edit my post to include the stronger assumption that X and Y are jointly normal. $\endgroup$ – 7Jack Jun 3 '16 at 19:59
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    $\begingroup$ "So only linear combinations of independent normal variables are guaranteed to be normal. If they are correlated this is no longer the case." is incorrect. Independence is in no way required. Linear combinations of random variables whose joint distribution is multivariate normal will follow the normal distribution (indeed, this is one way to define the multivariate normal distribution). Bivariate normal distribution is a special case of the multivariate normal distribution. $\endgroup$ – Matthew Gunn Jun 3 '16 at 20:40

If $X$ and $Y$ are correlated (univariate) normal random variables and $Z = AX+BY+C$, then the linearity of expectation and the bilinearity of the covariance function gives us that

\begin{align} E[Z] &= AE[X] + BE[Y] + C,\tag{1}\\ \operatorname{cov}(Z,X) &= \operatorname{cov}(AX+BY+C,X) = A\operatorname{var}(X) + B\operatorname{cov}(Y,X)\\ \operatorname{cov}(Z,Y) &= \operatorname{cov}(AX+BY+C,Y) = B\operatorname{var}(Y) + A\operatorname{cov}(X,Y)\\ \operatorname{var}(Z) &= \operatorname{var}(AX+BY+C) \quad = A^2\operatorname{var}(X) + B^2\operatorname{var}(Y) + 2AB \operatorname{cov}(X,Y), \tag{2}\\ \end{align} but it is not necessarily true that $Z$ is a normal (a.k.a Gaussian) random variable. That $X$ and $Y$ are jointly normal random variables is sufficient to assert that $Z = AX+BY+C$ is a normal random variable. Note that $X$ and $Y$ are not required to be independent; they can be correlated as long as they are jointly normal. For examples of normal random variables $X$ and $Y$ that are not jointly normal and yet their sum $X+Y$ is normal, see the answers to Is joint normality a necessary condition for the sum of normal random variables to be normal?. As pointed out at the end of my own answer there, joint normality means that all linear combinations $aX+bY$ are normal, whereas in the special case being discussed there, only one linear combination $X+Y$ of non-jointly normal random variables is proven to be normal; most other linear combinations are not normal.

More generally, if $X$ and $Y$ are (column) $n$-vector random variables with $n\times n$ covariance matrices $\Sigma_{X,X}$, $\Sigma_{Y,Y}$, and $n\times n$ crosscovariance matrix $\Sigma_{X,Y}$, $A$ and $B$ are $m\times n$ nonrandom matrices, and $Z$ and $C$ (column) $m$-vectors, then it is indeed true that \begin{align} E[Z] &= AE[X] + BE[Y] + C &\quad \scriptstyle{\text{compare with } (1)}\\ \Sigma_{Z,Z} &= A\Sigma_{X,X}A^T + B\Sigma_{Y,Y}B^T +2A\Sigma_{X,Y}B^T &\quad \scriptstyle{\text{compare with } (2)}\\ \end{align} but, as in the univariate case, it is not necessarily true that $Z$ is a normal vector (in the sense that the $m$ components $Z_i$ are jointly normal random variables). Once again, joint normality of $(X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n)$ suffices to allow the assertion that $Z$ is a normal random vector.

  • $\begingroup$ Great answer. The formulas for $cov(Z,X)$ and $cov(Z,Y)$ were what I was looking for. They give the same values that I obtained for covariance by doing Monte Carlo simulation in Matlab and recording sample covariance. $\endgroup$ – 7Jack Jun 4 '16 at 8:29

Covariance matrix $\Sigma_{xy}$ can be written: $$ \Sigma_{xy} = \left[ \begin{array}{cc} c_{xx} & c_{xy} \\ c_{yx} & c_{yy} \end{array} \right] $$

Where $c_{ab}$ denotes the covariance between $a$ and $b$. (Note symmetric so $c_{ab} = c_{ba}$.)

The covariance matrix for $X$,$Y$, and $Z$ would be: $$ \Sigma_{xyz} = \left[ \begin{array}{ccc} c_{xx} & c_{xy} & c_{xz} \\ c_{yx} & c_{yy} & c_{yz} \\ c_{zx} & c_{zy} & c_{zz} \\ \end{array} \right] $$

You need to find the additional terms:

$$\begin{align*} c_{zx} &= E[(Z - E[z])(X - E[x])] = \quad ?\\ c_{zy} &= E[(Z - E[z])(Y - E[y])] = \quad ?\\ c_{zz} &= E[(Z - E[z])^2] = \quad ?\\ \end{align*}$$


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