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.
For every $x_1\lt1$, the likelihood $L(\theta;x_1)=f(x_1;\theta)$ is an increasing function of $\theta$ hence, assuming that $\theta$ is rectricted to some interval $[0,\theta_*]$, one gets $\hat\theta_{\text{MLE}}(x_1)=\theta_*$. Likewise, for every $x_1\gt1$, $\hat\theta_{\text{MLE}}(x_1)=0$.