Currently attempting to better understand Maximum Likelihood Estimators with a sample problem that is a bit outside of my level of understanding. Given the observation $x$, the purpose is to find the MLE of $p$.
Starting with the probability function as it given:
$f(x) = (p) \cdot (1 - p)^{x-1}$
Deriving this, yields:
$f(d/dx) = \ln(1-p) \cdot (1-p)^{x-1} \cdot (p)$
Thinking about MLEs, the goal now would be to solve for when this derivative equates to $0$. Do I solve for $p$ or do I set $p$ to $0$? I'm not quite sure what the correct approach here is. The experiment represented by the probability function f(x) is run until it succeeds, at which point it stops.
The suggested response would be in the form of as an equation, which is something I have not yet encountered.
Any assistance would be quite appreciated.
You want to differentiate $f$ with respect to $p$, because you want to maximize the distribution $f$ with respect to the choice of parameter $p$.
The derivative would be $$f'(p)=(1-p)^{x-1}-p(1-p)^{x-2}(x-1)$$ So if you set that to zero and solve for $p$, you get $p=\frac{1}{x}$, or $p=1$. You should be able to show that $p=\frac{1}{x}$ gives the maximum.