Suppose I have $\hat{\mu}, b \in \mathbb{R}^d, B \in \mathbb{R}^{d \times d} $.
And sample $\tilde{\mu} \sim \mathcal{N}\left(\hat{\mu}, B^{-1}\right)$ (from multivariate normal distribution).
Then is it true that $b^{T}\tilde{\mu}$ is statistically equal to $a \sim \mathcal{N}\left(b^{T}\hat{\mu}, b^{T}B^{-1}b\right)$?
How can I prove it? and if it is true, then what happen when $a \sim \mathcal{N}\left(b^{T}\hat{\mu}, \alpha \ (b^{T}B^{-1}b)\right)$ ? ($\alpha \in \mathbb{R}$)
Is it still true that $b^{T}\tilde{\mu}$ is statistically equal to $a \sim \mathcal{N}\left(b^{T}\hat{\mu}, \alpha \ (b^{T}B^{-1}b)\right)$ in case of $\tilde{\mu} \sim \mathcal{N}\left(\hat{\mu}, \alpha B^{-1}\right)$ (or should be $\tilde{\mu} \sim \mathcal{N}\left(\hat{\mu}, \sqrt{\alpha} B^{-1}\right)$)?
Sorry for someone who see the question is obvious, but it will be really appreciate if I get any source or answer for understanding that question.