The second version of the Box$-$Muller algorithm.

$$ Generate U$1,$ U$2$ U$[1,1]$ until S $=$ U$21+$ U$22<=1.$

$$ Define Z $=$q$2$ log$($S$)/$S and take X$1=$ ZU$1$ and X$2=$ ZU$2.$

a$.$ Show that X$1,$ X$2$ generated via the above algorithm are iid from $$ N$(0,1).$

b$.$ Derive the average number of generations in step $1$ in the above algorithm in order to get $1000$ standard normal realizations.

c$.$ Examine the effect of not constraining $($U$1,$ U$2)$ to the unit circle for this algorithm.