Problem $1.$ Daily random demand D for newspapers at a newspaper stand has a Poisson distri$-$

bution with parameter $\backslash$lambda $>0.$ That is$,$

P$($D $=$ d$)=$ e$\backslash$lambda $\backslash$lambda d

d$!,$ for d $=0,1,2,....$

A Newsvendor has observed that the daily demands for newspapers over the past $8$ days were

$10,0,14,8,9,13,17,9.$ The Newsvendor utilizes this data to develop an MLE for $\backslash$lambda $,$ and then uses

this MLE to make order quantity decisions. Suppose the overage and underage costs are h $=15$

cents and p $=60$ cents. Decide the optimal order quantity Q$.$ Calculate an upper bound on the

probability that there will be a stock out, that is$,$ an upper bound on P$($D $>$ Q$).$