Problem $8\mathrm{.}5.$ Onder the usual assumption, if we further assume that $x_1$ has a density, then it can be shownthat

$(u)$~~$C{e}^{-Ru},$ for large $u$

in the sense that $\underset{u}{lim}(u)e^{Ru}=C,$ where

$C$:$=\frac{c-E[x_1]}{E[x_1{e}^{R{x}_1}]-c}$

This result is known as Cramer's asymptotic formula.

Show that if for some $>0$ we have that $P(x_{1}x)=1-e^{-x}$ for all $x0,$ then

$(u)=C{e}^{-Ru},$ for all $u>0$

Use Cramer's asymptotic formula in order to approximate the ruin probability $(u)$ from Problem