I am little confused regarding finding expectation of vector multiplied matrix. X is vector having mean $m_x \in \mathbb{R}^{N\times 1}$ and variance vector $s_x \in \mathbb{R}^{N\times N}$. $\phi$ is a vector with mean $\mu_\phi \in \mathbb{R}^{N\times 1}$ and $\Sigma_\phi \in \mathbb{R}^{N\times N}$. Now I want to find the mean of the expression $X^T \phi^T \phi X$.

Can anybody verify me in finding the closed form expression of this?

My way:

$ X^T \phi^T \phi X = \text{trace}((X^T X)(\phi \phi^T))$

So, $\mathbb{E}(X^T \phi^T \phi X) = \text{trace}(\mathbb{E}(X^T X)\mathbb{E}(\phi \phi^T)) = \text{trace}((\mu_\phi^T\mu_\phi + \Sigma_\phi)(m_x m_x^T + s_x))$.

Is this OK?

  • 1
    $\begingroup$ I'm curious about your use of the trace, given that $X^\prime X$ and $\phi^\prime\phi$ are both numbers. Therefore $$X^\prime \phi^\prime \phi X=X^\prime X\ \phi^\prime \phi$$(a product of two random numbers), following immediately from basic properties of scalar multiplication of matrices by the number $\phi^\prime\phi$. Employing the trace thereby suggests there may be some mismatch between the stated dimensions of $X$ and $\phi$ and their actual dimensions. Please note that in any case you must assume $X$ and $\phi$ are independent in order to complete your calculation. $\endgroup$
    – whuber
    May 30 '17 at 19:51
  • $\begingroup$ Here actually my objective is to find the expectation of $X^T \phi^T \phi X$ in terms of mean and variances of $X$ and $\phi$. $\endgroup$ May 30 '17 at 19:56
  • $\begingroup$ Right, that's what you wrote. Assuming you meant what you wrote about the dimensions of the vectors, and that the vectors are independent, your question reduces to two much simpler sub-problems: (1) to show that $E(AB)=E(A)E(B)$ for independent random variables $A$ and $B$ and (2) to show that the expectation of the square is the square of the mean plus the variance. But (1) is obvious and (2) results immediately from the definition of the covariance matrix--and both of those sub-problems have been addressed directly in other threads on this site. $\endgroup$
    – whuber
    May 30 '17 at 20:00

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