Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbf{x}$.

share|improve this question
    
I am putting no constraints on the joint distribution. Would it simplify matters to do so? –  DaemonMaker Nov 16 '12 at 16:09
add comment

2 Answers

up vote 2 down vote accepted

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.

share|improve this answer
    
What about the case when $\mathbf{x}$ and $\mathbf{y}$ are correlated? –  DaemonMaker Nov 16 '12 at 18:20
    
$C$ 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. –  Dilip Sarwate Nov 16 '12 at 19:29
    
+1. The last double sum is the trace of the matrix $AC^T$. –  Did Nov 17 '12 at 15:21
add comment

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

share|improve this answer
    
Thanks for the information. I updated my question to clarify. –  DaemonMaker Nov 16 '12 at 18:23
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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