Working through this given problem on maximum likelihood estimation (MLE). The density is given as
$$f(x \mid \theta) = \frac{1}{2\theta} e^{-|x|/\theta}, -\infty < x <\infty $$
I get
$$L(\theta)= \frac{1}{(2\theta)^{n}} e^{- \frac{1}{2\theta} \sum\limits_{i=1}^{n}|X_i|}$$
Applying the ln step
$$\ln L(\theta)= -n\ln(2\theta) - \frac{1}{\theta}\sum\limits_{i=1}^{n}|X_i|$$
Taking the derivative
$$\frac{d}{d\theta} = \frac{-n}{\theta} \frac{\sum\limits_{i=1}^{n}|X_i|}{\theta^{2}}$$
Setting the derivative to 0 and solving for $\theta$
$$\frac{-n}{\theta} \frac{\sum\limits_{i=1}^{n}|X_i|}{\theta^{2}} = 0 $$ $$\hat{\theta}= \frac{\sum\limits_{i=1}^{n}|X_i|}{n}$$
Is this correct or have i done a something wrong and im getting the wrong mle