Let $\{x_n{\}}_{nin{N}_o}$ be a simple symmetric random walk, and let $\{Y_n{\}}_{nin{N}_o}$ be a random process

whose value at time $nin{N}_O$ is equal to the amount of time$($number of steps, including

possibly $n)$ the process $\{x_n{\}}_{nin{N}_0}$ has spent above $0,$ i$.$e$.,$ in the set $\{1,2,$dots$\}.$ Then,

choose the correct answer

$($a$)\{Y_n{\}}_{nin{N}_0}$ is a Markov process

$($b$)Y_n$ is a function of $x_n$ for each ninN.

$($c$)x_n$ is a function of $Y_n$ for each ninN.

$($d$)Y_n$ is a stopping time for each $nin{N}_0.$

$($e$)$ None of the above