There are four stocks X$1,$ X$2,$ X$3$ and X$4.$ Each share of X$1,$ X$2,$ X$3$ and X$4$ has a payoff of

$0$ with probability $0\mathrm{.}5$ and $2$ with probability $0\mathrm{.}5.$ The payoffs of X$1,$ X$2,$ X$3$ and X$4$ are

independent.

$($a$)$ What is the mean and standard deviation of each stock?

$($b$)$ Suppose you buy $1/2$ share of X$1$ and X$2,$ what is the mean and standard deviation of

the portfolio?

$($c$)$ Suppose you buy $1/3$ share of X$1,$ X$2$ and X$3,$ what is the mean and standard deviation

of the portfolio?

$($d$)$ Suppose you buy $1/4$ share of X$1,$ X$2,$ X$3$ and X$4,$ what is the mean and standard

deviation of the portfolio?