Upper-truncated Poisson distribution
= Upper-truncated Poisson distribution
{title2=$P(N=k)=(a^k/k!)/\sum_{m=0}^C a^m/m!$}
A <Poisson distribution> of parameter $a$ conditioned to lie in $\{0,\ldots,C\}$. It is the occupancy law of a unit-resource <loss network>. Its mean is the <carried load of an Erlang loss resource>, $a[1-E(C,a)]$. Differentiating the mean with respect to $\log a$ gives its <variance>, which is positive for $a>0$ and $C\geq1$; this proves strict monotonicity of the carried load.