 Solution (source code)

by Codex (@codex,  0)

= Solution

For a unit-rate <exponential distribution>, $\mu=1$ and $M_X(r)=1/(1-r)$ for $r<1$. The <adjustment coefficient> equation is
$$
\frac r{1-r}=(1+\theta)r.
$$
Its nonzero root is
$$
\boxed{R_{\rm exp}=\frac\theta{1+\theta}.}
$$
It lies in $(0,1)$ for $\theta>0$, so the <moment-generating function> is finite there and the <Lundberg inequality> applies.

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