Example 9.1, p.463
$$ \ht\equiv\amt\cases{\lfdnp&X\text{ discrete}\\\lfcnp&X\text{ continuous}} $$
In this example, we have
$$ \ht=\amt\lfc{X}{x}{\theta}=\amt\cases{\frac1\theta&0\leq x\leq\theta\leq1\\0&\text{otherwise}}=x $$
The last equality follows because $\lfc{X}{x}{\theta}$ is decreasing on $[x,1]$ and is zero otherwise.
Example 9.2, p.463
Claim: “It follows that $\frac{k}{n}$ is also the ML estimate of $\theta$”. This follows because of the paragraph before Example 9.1: “Thus, if we view $\lfd{X}{x}{\theta}$ as a conditional PMF, we may interpret ML estimation as MAP estimation with a flat prior.” Then
$$ \lfd{X}{k}{\theta}=\cp{X=k}{\Theta=\theta} $$
The equality holds becausee $\Theta$ was assumed to be flat in example 8.8.
Claim: “so that the ML estimator is”
$$ \htb_n=\frac{X_1+\dots+X_n}n $$
This follows because of p.460: “Given observations $X=(X_l…..X_n)$. an estimator is a random variable of the form $\htb=g(X)$, for some function $g$. We use the term estimate to refer to an actual realized value of $\htb$.”
Claim: “It is also consistent, because $\htb_n$ converges to $\theta$ in probability, by the weak law of large numbers.” Note that
$$ \Ewrt{\theta}{X_i}=\lfd{X_i}{X_i=1}{\theta}=\cp{X_i=1}{\Theta=\theta}=\cptone{toss i is heads}{\Theta=\theta}=\theta $$
Claim: “This estimator is unbiased.”
$$ \Ewrt{\theta}{\htb_n}=\EBwrt{\theta}{\frac{X_1+\dots+X_n}n}=\frac1n\sum_{i=1}^n\Ewrt{\theta}{X_i}=\frac1n\sum_{i=1}^n\theta=\frac1nn\theta=\theta $$
p.464-465
From the top of p.465, we have
$$ m_n=\frac1n\sum_{i=1}^nx_i $$
Then
$$ \sum_{i=1}^n(x_i-m_n)(m_n-\mu)=(m_n-\mu)\sum_{i=1}^n(x_i-m_n)=0 $$
since
$$ \sum_{i=1}^n(x_i-m_n)=\sum_{i=1}^nx_i-\sum_{i=1}^nm_n=\sum_{i=1}^nx_i-nm_n=0 $$
since
$$ \sum_{i=1}^nx_i=n\frac1n\sum_{i=1}^nx_i=nm_n $$