Let $X$ be a random variable characterized by the following density function:
$f(x; \theta) = ((\theta + x) / (\theta + 1)) * exp (-x)$, if $x >= 0$
$f(x; \theta) = 0$, if $x < 0$
Assuming that 0 <= theta <= 4, determine a maximum likelihood estimate of the parameter theta based on the realization sample x1 = 1/2.
This is my partial solution, but I cannot do the first derivative...
Thank you for considering my request.