Maximum Likelihood Estimation of iid rvs

Question

Let $X_1,…,X_$ be iid rv’s with pmf given by $_(_)=(1−)^{_−1}$,with $x_i \in [1,2,…], \theta \in (0,1)$ with $_[_]=\frac{1}{}$

How would I calculate the MLE of $\theta$?

Answer 1

Let's consider the maximum likelihood function:
$L(\theta;\mathbf{x})=\theta^n\cdot{(1-\theta)^{(\sum_{i=1}^nx_i)-n}}$.
Now we derive the log-likelihood function and the score function:
$\mathcal{l}(\theta,\mathbf{x})=n\cdot{log(\theta)}+(\sum_{i=1}^nx_i-n)\cdot{log(1-\theta)}$,
$\mathcal{l}'(\theta,\mathbf{x})={n\over{\theta}}+{(n-\sum_{i=1}^nx_i)\over{1-\theta}}$.
Solving $\mathcal{l}'(\theta,\mathbf{x})>0$ you find that the MLE is given by:
$\hat\theta(\mathbf{x})={n\over{\sum_{i=1}^n}x_i}$