Just a small query: Q5. says find the MLE for theta and its mentioned that theta is > 1 . However the soln we get for MLE of theta is less than 1. So in this situation what do we do?
Generally speaking, if the parameter you're estimating is only valid in a certain range, you would take the maximum value that falls within that range (this is just the maximum likelihood within the range, after all). So in your case, it sounds like theta will be 1 (but check) or, strictly speaking, close to 1.
