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:
$L\left( \theta \right) = \theta ^{4}e^{-8.65\theta }$
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:
$\dfrac{dL\left( \theta \right) }{d\theta }=0$
Having calculated the derivative I obtained:
$\dfrac{dL\left( \theta \right) }{d\theta } = \theta ^{3}\left( 4-8.65\theta \right) \cdot e^{-8.65\theta }$
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?
It has a maximum at $\theta\approx 0.4624$ even if it is just $L(0.46)\approx 0.0008$
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