X is exponential RV with unknown parameter lambda, which we want to find.
Sample X=x
$ f_X(x;\lambda)=\lambda e^{-\lambda x} $
Therefore:
$ \lambda^{hat}_{ML}=max(\lambda e^{-\lambda x}) $
$ \frac{\delta}{\delta\lambda}(\lambda e^{-\lambda x})=e^{-\lambda x}-\lambda e^{-\lambda x}=0 $
Solving for lambda gives us:
$ \lambda^{hat}_{ML}=\frac{1}{x} $
