Let $X_1, \ldots, X_n$ be a random sample from the exponential distribution $\exp(\lambda)$. Let $$M_n=\max\{X_1, \ldots, X_n\}$$ with probability density function $$g_{M_n}(x)=n\lambda e^{-\lambda x}(1-e^{-\lambda x})^{(n-1)}, \qquad x>0$$
Q1. If $M_n$ is the only information that you have from the sample, find a maximum likelihood estimator (mle) $\hat{\lambda}_n$ of $\lambda$.
Q2. Using $(1+x)^n>1+nx$ prove that $\hat{\lambda}_n$ is consistent, i.e. that $P(| \hat{\lambda}_n-\lambda|>\epsilon)\longrightarrow0$, for $n\rightarrow \infty$
Thanks.