Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / M / M5 / a / vi

The remainder after the first child, $X_2$, is uniform on $0,\dots ,N-1$. Given $X_2=j\ge 1$, the second child takes an average of $(j+1)/2$ stickers; when $j=0$ the amount is zero. Hence, for $N\ge 1$,The expression gives zero for $N=1$, as expected.