Solution
= Solution
Write $p_j=\mathbb P(X=j)$, with $\sum_{j=0}^Np_j=1$. The <expected value>, <variance>, and strict lower-tail <probability> are
$$
\boxed{\mu=\sum_{j=0}^Njp_j,\qquad
\sigma^2=\sum_{j=0}^N(j-\mu)^2p_j
=\sum_{j=0}^Nj^2p_j-\mu^2,\qquad
\mathbb P(X<Y)=\sum_{j=0}^{Y-1}p_j.}
$$
The upper summation limit $Y-1$ is essential because the inequality is strict and the variable is integer-valued.