Atlas/Statistics Inference/Series/Series 6 .md

Consider the binomial distribution of the number k of successes in n independent trials:

\sum_{i=1}^{n} X_i \sim \text{Binomial}(n, p)

The log-likelihood function is:

\log L(p) = \log \binom{n}{k} + k \log(p) + (n-k) \log(1-p)

Part 1: Plot Log L(p)

Given n=100 and k=10 !

Part 2: Compute the MLE

Given: \sum_{i=1}^{n} x_i = 15 with n = 100 trials

Finding the MLE

To find the maximum likelihood estimator, we take the derivative of the log-likelihood with respect to p and set it equal to zero:

\frac{d}{dp} \log L(p) = \frac{k}{p} - \frac{n-k}{1-p} = 0

Solving for p:

\frac{k}{p} = \frac{n-k}{1-p} k(1-p) = p(n-k) k - kp = pn - pk k = pn \hat{p}_{MLE} = \frac{k}{n}

!