Q$2.$ To test $H_0$:$M=M_0$ against a suitable alternative hypothesis, where $M$ is the

median of a symmetric continuous distribution and $M_0$ is some given number

Wilcoxon proposed a test statistic given by

$T^{**}={\displaystyle \underset{i=1}{\overset{n}{}}}{z}_i$rank$(|D_i|),$ while $z_i=\{\begin{array}{ccc}1& if& D_i>0\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

where $D_i=x_i-M_0$

$($a$)$ Regarding $Z_1,Z_2,dots{Z}_n$ as iid Bernoulli random variables derive

$E\{\frac{T^{+}}{H_0}\}$ and Var$\{\frac{T^{+}}{H_0}\}$