I'm struggling with a question worded as follows:
Suppose that there are $N$ buses in a town. These numbers are numbered sequentially from $1$ to $N$. Your prior distribution of $N$ is given by a geometric distribution with mean 100. i.e.
$$ p(N) = 0.01(0.99)^{N-1} $$
If you see a bus at random and it is numbered 203, what is the posterior distribution for $N$?
I'm struggling to set up the likelihood function in order to find the posterior. Thanks
The distribution for your data given $N$ is uniform, so the pmf of $X|N$ is given by $1_{N\geq x}\frac{1}{N}$. This gives a posterior distribution proportional to $1_{N\geq 203}\frac{1}{N}(0.99)^{N-1}$. This looks like the logarithmic distribution, but not quite. In fact, if $Y$ is logarithmic, your posterior is the distribution of $Y|Y\geq 203$.