I was reading this post where the Markov Chain is represented as:
Can someone please explain how the mean and the variance can be calculated from the above equation?
PS: I know about Gaussian, and all the related equations but somehow I am not able to figure out the math
The (univariate) Gaussian distribution with mean $\mu$ and variance $\sigma^2$ has density proportional to $$ \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) =\exp\left(-\frac{(x^2-2\mu x+\mu^2)}{2\sigma^2}\right).\tag1 $$ If you have a univariate Gaussian distribution whose density is proportional to $$\exp\left(-\frac12(ax^2 -bx + c)\right),\tag2$$ then you obtain the mean and variance of your distribution by matching coefficients on $x^2$ and $x$ between (1) and (2). This gives the system of equations $$ -\frac1{2\sigma^2}=-\frac a2\qquad\text{and}\qquad \frac{\mu}{\sigma^2}=\frac b2, $$ which you can solve for $$\sigma^2=\frac1a\qquad\text{and}\qquad\mu=\frac b{2a}.$$ In your example this recipe will give you the conditional distribution for $x_{t-1}$ given $x_t$ and $x_0$.