The operations pop, init are in an obvious way in $O(1)$ in any scenario, so the average runtime will be

also in $O(1).[$Marking: $5$ marks for each of pop, init, correctly indeitified as $O(1)]$

Now push analysis. Assume we have n different elements $($comparable$,$ so they can be sorted in increasing

order$).$

In order to push an element in $k$ steps, we need to remove from top of the stack $k$ elements greater than

the element being inserted. Since we assume uniform distribution, each of the $k$ elements already in the

stack, are picked with following probabilities:

First element $1/$n Second $1/($n$-1)...$ k$-$th element $1/($n$-$k$).$

For them to occur together, probability is $\frac{1}{n(n-1)dots(n-k)}\frac{2}{n^2}$ for any $n2$ and precisely $\frac{1}{n}$ for $n=1.$

Hence $E(t=k)\frac{1}{n}+2\frac{2}{n^2}+$cdots$+2\frac{k}{n^2}\frac{1}{n}+2\frac{k^2}{n^2}3,$ showing averahe runtime is in $O(1).$

$[$marking: $5$ marks for a correct argument, $3$ marks for an approximately correct argument, $0$ otherwise$]$