MLE of a Bernoulli Parameter

Write the likelihood as a function of $p$ and take its logarithm.

Differentiate the log-likelihood with respect to $p$ and set it to zero.

1. 1. Write the log-likelihood

   The likelihood is $p^k(1-p{)}^{n-k}$, so the log-likelihood is $k\ln p+(n-k)\ln (1-p)$.

2. 2. Maximize

   Setting the derivative to zero, $\frac{k}{p}-\frac{n-k}{1-p}=0$, and solving gives the sample proportion.

   $$\hat{p}=\frac{k}{n}$$

Answer: k/n