Help! A question of MLE

#3
There is no theta in that density. :) Be careful with what you write!

(But is it not pretty obvious? What happens if you take the log?)
I wrongly wrote lamda in the place of theta, a little error. But the question is that I don't know how to tackle with the absolute value of (x-θ) after taking the log and summing up
 

Dason

Ambassador to the humans
#4
Can you show us what you have so far.

And completely unrelated... What properties do you know about the sample median?
 
#10
Have you been tasked with or at least told that the median minimizes the mean absolute deviation?
The first time I heard such a conclusion,but it is a good hint that we only need to prove that the minimum of the the mean absolute deviation is the median. Case closed then