Likelihood function:

$$L(\theta )=L(\theta ;x_1,x_2,...,x_n)=\prod _{i=1}^{n}f(x_i;\theta )=f(x_1;\theta )\cdot ...\cdot f(x_n;\theta )$$

It is often easier to consider

$$\text{ln}L(\theta )=\sum _{i=1}^n\text{ln}f(x_i;\theta )$$

Maximum Likelihood Estimator:

$$\hat{\theta }=\text{arg}maxL(\theta )=\text{arg}max\text{ln}L(\theta )$$

Method of Moments:

$$E(X)=h(\theta )\text{Set}\overline{X}=h(\tilde{\theta })\text{Solve for }\tilde{\theta }$$

Consider a single observation $X$ of a Binomial random variable with $n$ trials and probability of success $p$. That is,

$$P(X=k)=(\genfrac{}{}{0ex}{}{n}{k})p^k(1-p{)}^{n-k},k=0,1,...,n$$

Obtain the method of moments estimator of $p,\tilde{p}$

Binomial: $E(X)=np$

$X=n\tilde{p}\Rightarrow \tilde{p}=\frac{X}{n}$

Obtain the maximum likelihood estimator of $p,\hat{p}$

$$L(p)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}$$

$$\text{ln}L(p)=\text{ln}(\genfrac{}{}{0ex}{}{n}{x})+x\text{ ln}p+(n-x)\text{ln}(1-p)$$

$$\frac{d}{dp}\text{ln}L(p)=\frac{x}{p}-\frac{n-x}{1-p}=\frac{x-xp-np+xp}{p(1-p)}=\frac{x-np}{p(1-p)}$$