What is the expected value of $a^T X b$?

Question

Let $a, b\in\Bbb{R}^n$ and $X$ a random matrix with shape $n\times n$. Let $\Bbb{E}[X]=C$. Find $\Bbb{E}[a^T Xb]$. I did just that: $$ \Bbb{E}[a^TXb]=a^T\Bbb{E}[X]\ b=a^TCb $$ but I don't know if I did it right. How to prove or disprove this?

Answer 1

Note that $a^\intercal X b = \sum a_i b_j x_{i,j},$ so linearity of expectation shows that $$E(a^\intercal X b) = \sum a_i b_j E(x_{i,j}) = \sum a_i b_j c_{i,j} = a^\intercal C b.$$ Same calculation would apply for matrices $E(AXB) = AE(X)B.$

Answer 2

$a$ and $b$ are constants, the randomness here is only on $X$ and the thesis thus follows from the linearity of the expected value.

If you want to see it by computations, assume you have a probability space $(\Omega, \Sigma, P)$, where $\Omega \subset \mathbb{R}^{n \times n}$. Then we have: $$\mathbb{E}[a^T X b] = \int_{\Omega} a^T X b \ dP = a^T \left(\int_{\Omega} X \ dP\right) b = a^T \mathbb{E}[X] b \ ,$$ where the last step is a consequence of the linearity of the integral.