Discrete uniform mle

Question

Say we have a random sample $x_1, x_2, ..., x_n$ from the discrete uniform distribution, taken from integers ${\{1,2,...,N}\}$, then I want to find the mle of $N$. Now, through an indicator function approach, I found this to be $Y = max(x_1, x_2, ..., x_n)$. Now, I want to find out whether or not this estimator is unbiased or not and hence I need to find the expectation of $Y$. I found the probability mass function to be $\frac{z^n}{N^n} - \frac{(z-1)^{n}}{N^{n}}$, but I am stuck on trying to evaluate the expectation. I can't seem to manipulate the sum in to anything meaningful.

Answer 1

The CDF is $F_n(z)=\frac{z^n}{N^n}$. Since $M_n$, the maximum, is nonnegative,

$$E[M_n]=\sum_{z=0}^N (1-F(z))=N+1-\sum_{z=0}^Nz^n/N^n=N+1-\frac{1-(z/N)^{n+1}}{1-z/n}.$$