(b)

Given,

$\mu=\frac{\sum x_i}{n}$

$\sigma=\sqrt{\frac{\sum x_1^2}{n}-(\mu)^2}$

$\hat{\mu}=\frac{\Sigma y_i}{n}$

$=\frac{\frac{x_1+x_2}{2}+\frac{x_1+x_2}{2}+x_3+x_4+\ldots+x_n}{n}$

$\hat{\mu}=\frac{x_1+x_2+x_3 \ldots+x_n}{n}=\frac{\Sigma x_i}{n}=\mu$

$\sigma=\sqrt{\frac{\sum y_1^2}{n}-\left(\mu^{\prime}\right)^2}=\sqrt{\frac{\sum y_1^2}{n}-\mu^2}$

$\sum x_1^2=x_1^2+x_2^2+x_3^2+\ldots+x_n^2$

$\sum y_1^2=$

$\frac{\left(x_1+x_2\right)^2}{4}+\frac{\left(x_1+x_2\right)^2}{4}+x_3^2+x_4^2+\ldots+x_n^2$

$\Sigma x_1^y-\Sigma y_1^2=x_1^2+x_2^2-2 x_1 x_2=\left(x_1-x_2\right)^2 \geq 0$

$\sum x_1^2 \geq \Sigma y_1^2$