Binomial Distribution

Model

$$X\sim \text{Binomial}(n,p),p\in (0,1)$$

PMF

$$P(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x},x=0,1,\dots ,n$$

Expected Value

$$\mathbb{E}[X]=np$$

Variance

$$\mathrm{Var}(X)=np(1-p)$$

Likelihood (single observation)
Given observed $x$:

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

Ignoring constants:

$$L(p)\propto p^x(1-p{)}^{n-x}$$

Likelihood (i.i.d. sample)
Let $X_1,\dots ,X_m\sim \text{Binomial}(n,p)$ and

$$S=\sum _{i=1}^m{X}_i$$ $$L(p\mid x_1,\dots ,x_m)=\prod _{i=1}^m\left(\genfrac{}{}{0ex}{}{n}{x_i}\right)p^{x_i}(1-p{)}^{n-x_i}$$

Ignoring constants:

$$L(p)\propto p^S(1-p{)}^{mn-S}$$

Log-Likelihood

$$\ell (p)=S\log p+(mn-S)\log (1-p)+C$$