I have a signal X of a single value which goes through K circuits (k=1 to k=K). Each circuit adds some noise with a normal distribution with ($\mu_{k} $ $\sigma_{k}$) and each circuit is indepedant of each other of its own normal distribution.
I need a general expression that tells me the output of the signal Y. Is this suppose to be the expression?
$g(x,\mu_{k},\sigma_{k})=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-{\displaystyle {\frac{(x-\mu)^{2}}{2\sigma^{2}}}}}$
$Y=X+\Sigma_{k=1}^{K}\left(g(x,\mu_{k},\sigma_{k})\right)$
Can that summation of gaussians with independant mu and sigma be simplified? If so, is it possible to compute an expression for the MAP of X?
TLDR: Even a simple example would help me.
If I have X which follows a gaussian distribution, and K which follows a gaussian distribution. What is the MAP of X given Y and K?
Your expression of $Y$ is not correct; you can not add probability density functions to random variables. The right choice, according to your description, should be $$Y = X + \sum_{i = 1}^K Z_i,$$ where $Z_i$'s are i.i.d. Gaussian random variables independent of $X$, with $Z_i \sim \mathcal{N}(\mu_k, \sigma_k^2)$.
Let $ Z = \sum_{i = 1}^K Z_i$, then $Z$ is also a Gaussian random variable. What you need to do is to calculate the posterior probability $p_{X|Y}(x|y)$.