Bernoulli Likelihood

Model

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

PMF

$$P(X=x)=p^x(1-p{)}^{1-x},x\in \{0,1\}$$

Likelihood (single observation)

$$L(p\mid x)=p^x(1-p{)}^{1-x}$$

Likelihood (i.i.d. sample)
Let $X_1,\dots ,X_n\sim \text{Bernoulli}(p)$ and $k={\sum }_{i=1}^n{X}_i$.

$$L(p\mid x_1,\dots ,x_n)=p^k(1-p{)}^{n-k}$$

Log-Likelihood

$$\ell (p)=k\log p+(n-k)\log (1-p)$$