(Solved): Let \( X_{1}, \ldots, X_{n} \) be a random sample from a distribution \( X \) with density function ...

Let \( X_{1}, \ldots, X_{n} \) be a random sample from a distribution \( X \) with density function (that depends on \( (\lambda, \theta) \) ) \[ f(x ; \theta, \lambda)=\left\{\begin{array}{ll} \lambda e^{-\lambda(x-\theta)}, & x>\theta, \\ 0, & x \leq \theta, \end{array}\right. \] with \( \lambda>0 \) and \( \theta \in \mathbb{R} \). 1. Find the methods of moments estimator for \( (\lambda, \theta) \). 2. Find the corresponding estimate of \( (\lambda, \theta) \) when \( n=10 \) and \( x_{1}=3 \), \( x_{2}=0.5, x_{3}=2.5, x_{4}=2, x_{5}=5, x_{6}=3.5, x_{7}=10, x_{8}=9 \), \( x_{9}=18, x_{10}=1.5 \).