Without information about the previous measurement, a random variable $X_1$ has pdf given by a Log-Normal distribution with known mean $\mu$ and variance $\sigma$. If we are given that the autocorrelation coefficient for $R(t) = \alpha, t = 1$ is a, what is the pdf of $X_2$ for a given $x_1$?
How is this found more generally for a continuous variable of any pdf?