Consider the following algorithm for generating a

random sample $S$ of size $n$ from the set of integers $\{1,2,$dots,$N\},$ where $IS{I}_{i}Si-1Si{I}_{i}I{I}_{i}E[I]NnVar(I)$ Sample $(N,n)$ :

$I=0,$Slarr$\frac{O}{,}\frac{\frac{?}{?}}{S}is$ a set $of$ distinct integers, initially $an$ empty set

while $|S|$

Let I $be$ the number $of$ die rolls until $Sis$ returned. Also, let $I_{i}be$ the random variable which describes the

number $of$ rolls $it$ takes from the time the set $S$ has $i-1$ values $to$ the first time a new value $is$ added after that

$(i.e.,$ the set $S$ has i values$).$

Answer the following questions $(6$ points each$)$

$a0.$

Sample $(N,n)$ :

$I=0,$Slarr$\frac{O}{,}\frac{\frac{?}{?}}{S}is$ a set $of$ distinct integers, initially $an$ empty set

while $|S|$

Let I $be$ the number $of$ die rolls until $Sis$ returned. Also, let $I_{i}be$ the random variable which describes the

number $of$ rolls $it$ takes from the time the set $S$ has $i-1$ values $to$ the first time a new value $is$ added after that

$(i.e.,$ the set $S$ has i values$).$

Answer the following questions $(6$ points each$)$

$a$