A single inventory is controlled by a continuous review $($R$,$Q$)$ policy. The lead time demand is normally distributed with mean $20$ and standard deviation $5.$ The average inventory position is $30$ The service level S$1=0\mathrm{.}90.$

$1.$ Calculate S$2$

$2.$ Now instead the lead time demand follows an exponential distribution with mean $20.($all other assumptions are the same as before$)$ Calculate S$1$

$3.$Assume again that the lead time demand follows an exponential distribution with mean $20.$ Determine the value of R if we want to attain the same level of S$1$ as in the original case $(90\%)?$