Hi guys

I'm trying to solve this

I have this pdf

$\displaystyle f(x)=e^{-(x-a)} , x > a$

$\displaystyle f(x)=0 , x \leq a$

I must find the maximum likelihood estimator of the expected value

I did this

$\displaystyle f(x_1|a)...f(x_n|a)=e^{-(x_1-a)}...e^{-(x_n-a)}=e^{\sum_{i=1}^{n}-x_i}e^{na}$

The log of the last expression is

$\displaystyle {\sum_{i=1}^{n}-x_i}+na$

In its derivative the $\displaystyle a$ disappears

Without taking the log I end maximizing the same expression (the exponent of $\displaystyle e$)

How can I proceed?

I see that I can express the pdf in this way:

$\displaystyle 1e^{-1x}e^{a}$

This is the exponential distribution multiplied by a constant (but i know that it is defined for different values of x)

Is this a wrong path?

Thanks!