We have usual matrix linear regression model $Y=X \beta+\epsilon$ , where $E(\epsilon) = 0 $ and $\operatorname{Var}(\epsilon)=\sigma^2 I$ and $\hat{\beta}$ is the least squares estimator of the parameter $\beta$ . For $n$- dimensional nonzero fixed vector $a$ and some $m-$dimensional fixed vector $b$ we define $A=a^T \epsilon+b^T\beta$.
What is the conditional expectation $E(A\mid Y)$ in that case?
Since $A=a^T \epsilon+b^T\beta $, I would say that $E(A\mid Y)=A$, but I am not sure about it, because in continuation of this task I don't get the right answer if I use this.
Thanks for the help.
Sorry for posting this as an answer (can't comment). It seems something is not right here (or missing), since $$ \mathbb{E}[A \mid Y] = \mathbb{E}[a^T \varepsilon + b^T \beta \mid Y] = b^T \beta + \mathbb{E}[a^T \varepsilon \mid Y] $$
From the setup you provided it's not quite clear what the expression $\mathbb{E}[a^T \varepsilon \mid Y]$ stands for. I would guess that $\mathbb{E}[a^T \varepsilon \mid Y] = 0$ by treating $Y$ as an observations which are independent of assumed in the model noise $\varepsilon$ and obtaining $\mathbb{E}[A \mid Y] = b^T \beta$. Otherwise, we need to deal with the object $\mathbb{E} [a^T \varepsilon \mid \varepsilon]$.