I have a pdf given as,
$$ \begin{equation} \nonumber \mathcal{p_\lambda}(x) = \left\{ \begin{array}{l l} ae^{-\lambda (x - b)} & \quad x \geq b\\ 0 & \quad \textrm{if } x < b \end{array} \right. \end{equation} $$
How do I calculate its MLE?
I've tried doing it. And this is what I ended up with: $$ \lambda_{MLE} = \prod_{i = 1}^{n}f_X(x_i; \lambda) = \prod_{i = 1}^{n} \bigl\{{ae^{-\lambda(x_i - b)}}\bigr\} \\ L(\lambda) = ae^{-n\lambda((x_1 + x_2 + ... + x_n) - b)}\\ \log L(\lambda) = \log (a) - \lambda x log(e) + \lambda b \log(e) \\ \text{Taking derivative on both sides,} \\ \frac{d}{d\lambda} \{\log L(\lambda)\} = \frac{d}{d\lambda} \{\log (a) - \lambda x log(e) + \lambda b \log(e)\} \\ \Rightarrow b - \bar x \\ $$ Now I know that I have to equate the derivative to 0 to get MLE, but there's no $\lambda $ to get the $\lambda_{MLE}$. How shall I proceed? Am I doing something wrong? Please, help!