这是一份umass麻省大学 STAT 515作业代写的成功案例
To determine the expected value of a chi-squared random variable, note first that for a standard normal random variable $Z$,
$$
\begin{aligned}
1 &=\operatorname{Var}(Z) \
&=E\left[Z^{2}\right]-(E[Z])^{2} \
&=E\left[Z^{2}\right] \quad \text { since } E[Z]=0
\end{aligned}
$$
Hence, $E\left[Z^{2}\right]=1$ and so
$$
E\left[\sum_{i=1}^{n} Z_{i}^{2}\right]=\sum_{i=1}^{n} E\left[Z_{i}^{2}\right]=n
$$
The expected value of a chi-squared random variable is equal to its number of degrees of freedom.
Suppose now that we have a sample $X_{1}, \ldots, X_{n}$ from a normal population having mean $\mu$ and variance $\sigma^{2}$. Consider the sample variance $S^{2}$ defined by
$$
S^{2}=\frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}{n-1}
$$
STAT515 COURSE NOTES ：
If the population mean $\mu$ is known, then the appropriate estimator of the population variance $\sigma^{2}$ is
$$
\frac{\sum_{i=1}^{n}\left(X_{i}-\mu\right)^{2}}{n}
$$
If the population mean $\mu$ is unknown, then the appropriate estimator of the population variance $\sigma^{2}$ is
$$
S^{2}=\frac{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}{n-1}
$$
$S^{2}$ is an unbiased estimator of $\sigma^{2}$, that is,
$$
E\left[S^{2}\right]=\sigma^{2}
$$