这是一份GLA格拉斯哥大学STATS5021_1作业代写的成功案例
If every $y$ in the population is multiplied by a constant $a$, the expected value is also multiplied by $a$ :
$$
E(a y)=a E(y)=a \mu .
$$
The sample mean has a similar property. If $z_{i}=a y_{i}$ for $i=1,2, \ldots, n$, then
$$
\bar{z}=a \bar{y}
$$
The variance of the population is defined as $\operatorname{var}(y)=\sigma^{2}=E(y-\mu)^{2}$. This is the average squared deviation from the mean and is thus an indication of the extent to which the values of $y$ are spread or scattered. It can be shown that $\sigma^{2}=E\left(y^{2}\right)-\mu^{2}$.
The sample variance is defined as
$$
s^{2}=\frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}}{n-1}
$$
which can be shown to be equal to
$$
s^{2}=\frac{\sum_{i=1}^{n} y_{i}^{2}-n \bar{y}^{2}}{n-1} .
$$
STATS5021_1 COURSE NOTES ：
$$
\begin{aligned}
z_{i} &=a_{1} y_{i 1}+a_{2} y_{i 2}+\cdots+a_{p} y_{i p} \
&=\mathbf{a}^{\prime} \mathbf{y}{i}, \quad i=1,2, \ldots, n \end{aligned} $$ The sample mean of $z$ can be found either by averaging the $n$ values $z{1}=\mathbf{a}^{\prime} \mathbf{y}{1}, z{2}=$ $\mathbf{a}^{\prime} \mathbf{y}{2}, \ldots, z{n}=\mathbf{a}^{\prime} \mathbf{y}{n}$ or as a linear combination of $\overline{\mathbf{y}}$, the sample mean vector of $\mathbf{y}{1}$, $\mathbf{y}{2}, \ldots, \mathbf{y}{n}$ :
$$
\bar{z}=\frac{1}{n} \sum_{i=1}^{n} z_{i}=\mathbf{a}^{\prime} \bar{y}
$$