Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 18 / a / Solution

Write $p_j=\mathbb{P}(X=j)$, with ${\sum }_{j=0}^N{p}_j=1$. The expected value, variance, and strict lower-tail probability areThe upper summation limit $Y-1$ is essential because the inequality is strict and the variable is integer-valued.