There is a comet whose model show it is moving along a straight line
$$ x(t) = mt + b $$
I have the posterior predictive distribution of a comet's position, which is
$$ PPD(x) = p(x|t,d) ∝ \exp\left( -\frac{1}{2} \frac{[x-Mt-B]^2}{σ^2t^2 + ξ^2}\right) $$
where $x$ is the position of the comet at some future time $t$. $M$ and $σ^2$ are the mean and variance of orbital parameter $m$, and $B$ and $ξ^2$ are the mean and variance of orbital parameter $b$. How could I determine the most likely position of the comet at $t$ and the mean of the PPD as a function of $t$?
For the mean of the PPD, I thought the solution could be just
$$ \int t \exp\left( -\frac{1}{2} \frac{[x-Mt-B]^2}{σ^2t^2 + ξ^2}\right)dt$$
but I'm not sure if this is correct.