GBE PS2

Statistics Inference/Series/Series 6 .md

@@ -0,0 +1,34 @@
+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 $k=10$ 
+![[image-1.png]]
+
+## 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 $p$:
+
+$$\frac{k}{p} = \frac{n-k}{1-p}$$
+
+$$k(1-p) = p(n-k)$$
+
+$$k - kp = pn - pk$$
+
+$$k = pn$$
+
+$$\hat{p}_{MLE} = \frac{k}{n}$$
+![[image-2.png]]