= Adjustment coefficient for shape-two Erlang claims
{title2=$R=\frac{2k-1-\sqrt{1+4k}}{k\beta},\quad k=\frac{2c}{\lambda\beta}>2$}
For an <Erlang distribution> of shape two and rate $2/\beta$, the claim <expected value> is $\beta$ and its <moment-generating function> is $(1-\beta r/2)^{-2}$ for $r<2/\beta$. The <adjustment coefficient> equation is $\lambda[(1-\beta R/2)^{-2}-1]=cR$. With $z=\beta R/2$, divide out the zero root to get $k(1-z)^2=2-z$. The smaller quadratic root lies in $(0,1)$ when $c>\lambda\beta$; the larger root lies beyond the finite-transform domain and must be rejected. This proves the displayed coefficient.
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