I have to work out whether its weakly stationary or not
Xt = b$\displaystyle Z_0$ where b is a constant.
I know i have to work out that the mean first and make it finite and not dependent on t.
What does E[$\displaystyle Z_0$] = ????
Do you by any chance mean
$\displaystyle X_t$= b$\displaystyle X_t_-_1$ + $\displaystyle e_t$ where e is a zero mean independent identically distributed noise process with a variance?
If so $\displaystyle X_t_+_n$= $\displaystyle b^n$$\displaystyle X_t$ + a bunch of mean zero terms with variance equal to var(e)/(1-b^2). Since b^n converges to zero the series is stationary.
As you have it written $\displaystyle X_t$ is generated by repeated draws from an iid distribution and isnt really a time series.