Show the mean of a non-negative integer-valued variable is finite
Solution 1:
We have $$ \sum_{k=0}^{\infty} P\left(\frac XN > k\right)=\sum_{k=0}^{\infty} P\left(X >N k\right)\leqslant \sum_{k=0}^{\infty}p^k<\infty $$ hence $X/N$ has a finite expectation and so has $X$.