I would like to get some explanation or intuition behind the process of finding the maximum likelihood of an exponential distribution.
Given the following likelihood function:
$Lleft( theta right) = theta ^{4}e^{-8.65theta }$
I would like to find $theta$ for which the likelihood is maximised. I know I can achieve this by taking the derivative and solving the equation:
$dfrac{dLleft( theta right) }{dtheta }=0$
Having calculated the derivative I obtained:
$dfrac{dLleft( theta right) }{dtheta } = theta ^{3}left( 4-8.65theta right) cdot e^{-8.65theta }$
Solving it for $0$ yields ~ $0.46$.
When one takes a look at the plot of this function, one can see that this function doesn’t have the maximum in that place. Since it is $e^(-x)$, the max value goes to the inf.
What am I actually determining, if I say that likelihood is maximised at 0.46?
