LetX1, . . . ,Xn be a random sample from a uniform distribution on the interval (a, b).
How can I derive the maximum likelihood estimators for a and b?
If you look at the likelihood function, its derivative is always positive.
Hence it's always increasing. So the maximum is attained for the highest possible value for the parameter...
But if you have observations x1,...,xn, they're all between a and b.
So for any i, a<xi... this means that the MLE for a will be min(xi)
And xi<b... this means that the MLE for b can be... anything...