这是一份andrews圣安德鲁斯大学 MT2508作业代写的成功案例
(a) Show that the likelihood for model $M_{k}$ is given by
$$
l\left(\theta_{k}, k\right)=\prod_{j=0}^{k} \lambda_{j}^{d_{j}} e^{-\lambda_{j}\left(t_{j+1}-t_{j}\right)}
$$
where $d_{j}$ is the number of occurrences in interval $I_{j}, j=0,1, \ldots, k$.
(b) Show that the likelihood ratio between model $M_{k}$ with parameters $\theta_{k}^{\prime}$ and $\theta_{k}$ is given by
$$
\operatorname{lr}=\prod_{j=0}^{k}\left(\frac{\lambda_{j}^{\prime}}{\lambda_{j}}\right)^{d_{j}} e^{\left(\lambda_{j}-\lambda_{j}^{\prime}\right)\left(t_{j+1}-t_{j}\right)}
$$
if they differ only on the values of $\lambda(k)$ and
$$
l r_{t}=\prod_{j=0}^{k} \lambda_{j}^{d_{j}^{\prime}-d_{j}} e^{\lambda_{j}\left[\left(t_{j+1}-t_{j}\right)-\left(t_{j+1}^{\prime}-t_{j}^{\prime}\right)\right]}
$$
MT2508 COURSE NOTES ：
$x$ has normal distribution with mean $\mu$ and variance $\sigma^{2}$, denoted by $N\left(\mu, \sigma^{2}\right)$, if its density is
$$
f_{N}\left(x ; \mu, \sigma^{2}\right)=\frac{1}{\sqrt{2 \pi \sigma^{2}}} \exp \left{-\frac{1}{2 \sigma^{2}}(x-\mu)^{2}\right} .
$$
The standard normal distribution is obtained when $\mu=0$ and $\sigma^{2}=1$.
As will be seen in the next chapter, it is particularly advantageous in the Bayesian context to work with the reparametrization $\phi=1 / \sigma^{2}$. The parameter $\phi$, the inverse of the variance, is usually referred to as precision. For this parametrization, the density is
$$
f_{N}\left(x ; \mu, \phi^{-1}\right)=\frac{\phi^{1 / 2}}{\sqrt{2 \pi}} \exp \left{-\frac{\phi}{2}(x-\mu)^{2}\right}
$$