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:

$\lambda^{hat}_{ML}=\frac{1}{x}$

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