Combine normal prior with observation

Question

I have unobserved random variable $X$ from (multivariate) normal distribution. $X \sim N(\mu, \Sigma)$. I observe a value (vector) $Z$ which is noisy linear transformation of $X$. $$Z = H X + W$$ $W$ is random noise from (multivariate) normal distribution. $W \sim N(0, Q)$.

What is posterior conditional probability density function of $X$ after observing $Z$ and why?

Edit: $W$ and $X$ are independent.

I expected that the answer should be:

$X|Z \sim N(\mu+K (Z-H\mu), (I - K H)\Sigma)$ where $K = \Sigma H^T (Q+H \Sigma H^T)^{-1}$

I know this formula from Kalman filter. However I don't understand this formula at all. I think that the answer should be equivalent with this formula which I would like to understand.

Answer 1

$$f(X|Z) = \frac{f(Z|X)f(X)}{f(Z)}$$

$$f(Z|X) \sim \mathcal{N}(HX,Q)$$

$$f(X) \sim \mathcal{N}(\mu, \Sigma)$$

$$f(Z) \sim \mathcal{N}(H\mu,H\Sigma H^{T}+Q)$$

I assumed that $X$ and $W$ are independent. Note that $Z$ is sum of two normal random variables which will be a random variable. Take expectation on both sides to get the mean of $Z$ and similarly for variance.