I haven't done stats in a very long time and I was never much good at it. I've been trying to solve the following problem but I'm not having much luck:
Let $(x,y)\in\mathbb{R}^2$ be a Gaussian random variable, specified by the distributions $x\sim N(m',c')$ and $y|x\sim N(x,\gamma)$, $c',\gamma>0$. Show that $x|y\sim N(m,c)$ where
$\frac{1}{c}=\frac{1}{c'}+\frac{1}{\gamma}$, and
$\frac{m}{c}=\frac{m'}{c'}+\frac{y}{\gamma}$
Any help will be greatly appreciated.
This is an exercise with Bayes law https://en.wikipedia.org/wiki/Bayes%27_theorem. It is better to write the random variables as capitals $X,Y$. You know that $Y\sim N(m', c'+\gamma)$ since a sum of two independent Gaussians is Gaussian with the means and variances added. Thus you know the density $$f_Y(y)= \exp\Bigl( (y-m')^2/(2c'+2\gamma)\Bigr)/\Bigl(\sqrt{2\pi(c'+\gamma)}\Bigr) \, . $$
The joint density $f(x,y)$ is $$ f(x,y)=\exp\Bigl(-(x-m')^2/(2c')- (y-x)^2/2\gamma\Bigr)/\Bigl(2\pi\sqrt{\gamma c'}\Bigr) \, . $$
Finally, Bayes rule gives the conditional density $$ f(x|y)=f(x,y)/f_Y(y) $$ which is Gaussian with the parameters you specified.