这是一份warwick华威大学ST227-10 的成功案例

For a given $k$-tuple, $\left(x_{1}, x_{2}, \ldots, x_{k}\right)$, let
$$
\mathbf{x}{\text {obs }}^{2}=\sum{i=1}^{k} \frac{\left(x_{i}-n p_{i}\right)^{2}}{n p_{i}}
$$
be the observed value of Pearson’s statistic. Use the chi-square approximation to the distribution of $\mathbf{X}^{2}$ to compute $P\left(\mathbf{X}^{2} \geq \mathbf{X}{\mathrm{obs}}^{2}\right)$. Then the following hold: (i) If $P\left(\mathbf{X}^{2} \geq \mathbf{X}{\text {obs }}^{2}\right)>0.10$, the fit is judged to be good (the observed data are judged to be consistent with the multinomial model).
(ii) If $0.05<P\left(\mathbf{X}^{2} \geq \mathbf{x}{\mathrm{obs}}^{2}\right)<0.10$, the fit is judged to be fair (the observed data are judged to be marginally consistent with the model). (iii) If $P\left(\mathbf{X}^{2} \geq \mathbf{X}{\text {obs }}^{2}\right)<0.05$, the fit is judged to be poor (the observed data are judged to be not consistent with the model).

The probability $P\left(\mathbf{X}^{2} \geq \mathbf{x}_{\text {obs }}^{2}\right)$ is called the $p$ value of the test. The p value measures the strength of the evidence against the given multinomial model.

ST229-10 COURSE NOTES ：

The bias of the estimator $\widehat{\theta}$ is the difference between the expected value of the estimator and the true parameter:
$$
B \operatorname{BIS}(\widehat{\theta})=E(\widehat{\theta})-\theta
$$
If $E(\widehat{\theta})=\theta$, then $\widehat{\theta}$ is said to be an unbiased estimator of $\theta$; otherwise, $\widehat{\theta}$ is said to be a biased estimator of $\theta$.

For example, let $\bar{X}, S^{2}$, and $S$ be the sample mean, sample variance, and sample standard deviation of a random sample of size $n$ from a normal distribution. Since $E(\bar{X})=\mu, \bar{X}$ is an unbiased estimator of $\mu$. Since $E\left(S^{2}\right)=\sigma^{2}, S^{2}$ is an unbiased estimator of $\sigma^{2}$. Since
$$
E(S)=\sigma \sqrt{\frac{2}{n-1}} \frac{\Gamma(n / 2)}{\Gamma((n-1) / 2)} \neq \sigma
$$
$S$ is a biased estimator of $\sigma .$