$X$ is exponentially distributed $\varepsilon(\theta)$. Using the Method of Maximum likelihood find the best (marking?)of sample $n$ for parameter $\theta$ .Question its centeredness and existence. Now I think I do not know the exact english translations of these notions so I will explain using the ones I do.
First of all I found out the the "marking" I am looking for is $$\overline{\theta_n}=\frac{1}{\overline{X_n}}$$ using miximum likihood method where $\overline{X_n}$ is the sample mean. Now by centeredness of the "marking" I mean that the expectation $E(\overline{\theta_n})=\overline{\theta}$ and existence meaning that $\overline{\theta_n}$ converges in probability to $\theta...$
As far as I understand, you found the maximum-likelihood estimator of $\theta$, i.e. $$ \hat\theta_n=\frac1{\overline X_n}, $$ and now you want to show that it is unbiased and consistent.
We have that $$ \operatorname E\hat\theta_n=\frac n{n-1}\theta $$ (see here for a complete explanation how to calculate this expected value). Hence, $\hat\theta_n$ is not an unbiased estimator.
By the weak law of large numbers, $$ \overline X_n\to\frac1\theta $$ in probability as $n\to\infty$. By the continuous mapping theorem, $$ \hat\theta_n=\frac1{\overline X_n}\to\theta $$ in probability as $n\to\infty$. Hence, $\hat\theta_n$ is a consistent estimator.