Given two multivariate random variables $\mathbf{x} \sim N(\hat{\mathbf{x}}, Q)$, $\mathbf{y}\sim N(\hat{\mathbf{y}}, R)$ which are not independent but are correlated with covariance matrix $C$ and a constant matrix $A$. What is the expectation of $\mathbf{x}^T A \mathbf{y}$, i.e. $E[\mathbf{x}^T A \mathbf{y}]$? What if $\mathbf{x}$ and $\mathbf{y}$ are independent? Note that I'm using $\mathbf{x}^T$ to mean the transpose of $\mathbf{x}$.

  • $\begingroup$ I am putting no constraints on the joint distribution. Would it simplify matters to do so? $\endgroup$ – DaemonMaker Nov 16 '12 at 16:09

Edited in response to OP's edits of the question

$C$ is $n\times n$ covariance matrix of (column) $n$-vectors $\mathbf x$ and $\mathbf y$. The $i$-$j$-th entry in $C$ is $c_{i,j} = \text{cov}(X_i, Y_j)$. Now, $E[X_i]=\hat{x}_i$ and $E[Y_j]=\hat{y}_j$ so that $c_{i,j} = E[X_iY_j]-\hat{x}_i\hat{y}_j$. Then, $$\begin{align*} E[\mathbf x^TA\mathbf y] &= E\left[\sum_{i=1}^n\sum_{j=1}^n a_{i,j}X_iY_j\right]\\ &= \sum_{i=1}^n\sum_{j=1}^n a_{i,j}E[X_iY_j]\\ &= \sum_{i=1}^n\sum_{j=1}^n a_{i,j}(c_{i,j}+\hat{x}_i\hat{y}_j)\\ &= \hat{\bf x}\,^TA\hat{\bf y} + \sum_{i=1}^n\sum_{j=1}^n a_{i,j}c_{i,j}. \end{align*}$$ If $C = \mathbf 0$ is the all-zeroes matrix, then we get $$E[\mathbf x^TA\mathbf y] = \hat{\mathbf x}\,^TA\hat{\mathbf y} = E[\mathbf{x}]^TAE[\bf y]$$ as in Robert Israel's answer. (However, Robert has not as yet revised his answer in response to your edits that make the $X_i$ and $Y_j$ non-zero-mean random variables and so $E[\mathbf{x}]^TAE[\mathbf y]$ does not equal $0$ when $C = \mathbf 0$ as he says). Note that normality of $\mathbf x$ and/or $\mathbf y$ has nothing to do with the matter.

  • $\begingroup$ What about the case when $\mathbf{x}$ and $\mathbf{y}$ are correlated? $\endgroup$ – DaemonMaker Nov 16 '12 at 18:20
  • $\begingroup$ $C$ is the matrix from which you can read off the covariance of any component $X_i$ of $\mathbf x$ and any component $Y_j$ of $\mathbf y$. The correlation matrices $Q$ and $R$ specify only the covariances of $X_i$ and $X_j$ and $Y_i$ and $Y_j$ respectively, and are not relevant to the computation of $E[\mathbf x^TA\mathbf y]$ which depends on $C$ and not at all on what $Q$ and $R$ are. $\endgroup$ – Dilip Sarwate Nov 16 '12 at 19:29
  • $\begingroup$ +1. The last double sum is the trace of the matrix $AC^T$. $\endgroup$ – Did Nov 17 '12 at 15:21

Unless $\bf x$ and $\bf y$ are uncorrelated, you haven't given us enough information. If they are uncorrelated, $E[{\bf x}^T A {\bf y}] = E[{\bf x}]^T A E[{\bf y}] = 0$.

  • $\begingroup$ Thanks for the information. I updated my question to clarify. $\endgroup$ – DaemonMaker Nov 16 '12 at 18:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.