$1.$ Please let $Y={\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i,$ where $x_{i}i.i.d.$ for iinN with $E[x_i]={}_x,$

Var$(x)={}_x^2,$ and $NPoisson().$ Please calculate the quantities below.

$a-E[Y]$

$b-$ Var$(Y)$

$2.$Prof. $Dr.Gullu(and$ Halis$)$ help a company manage their inventory

system. Prof. $Dr.Gullu$ has shown Halis how $to$ perform Stochastic Dynamic Programming, and $asan$ exercise asked him $to$ calculate the optimal

total expected profit $of$ the inventory system over $3$ time periods. The information $we$ have are $as$ follows: the inventory capacity $is3,$ and the demand

may take values $0,1$ and $2$ with probabilities $0\mathrm{.}25,0\mathrm{.}5$ and $0\mathrm{.}25$ respectively.

Each unit $is$ sold for $$8,$ and costs $$2.$ Placing $an$ order has a fixed cost $of$ $$4,$

and $at$ the beginning $of$ the period placed orders are received immediately,

$i.e.$ with $no$ lead$-$time. $At$ the end $of$ each period a cost $of$ $$1$ per unit per

period $is$ incurred $if$ any units remain unsold. There $isno$ back$-$ordering

possible, $i.e.if$ demand $is$ not satisfied $itis$ lost with $no$ penalty cost. $Ifwe$

start with $an$ inventory $of0$ units, and any products remaining $at$ hand $at$

the end $of$ period $3$ have $no$ salvage value, please help Halis calculate the

optimal total expected profit $of$ the inventory system over $3$ time periods.