In this paper, the problem of inverse optimal control for finite-horizon discrete-time Linear Quadratic Regulators (LQRs) is considered. The goal of the inverse optimal control problem is to recover the corresponding objective function by the noisy observations. We consider the problem of inverse optimal control in two scenarios: 1) the distributions of the initial state and the observation noise are unknown, yet the exact observations on the initial states and the noisy observations on system output are available; 2) the exact observations on the initial states are not available, yet the observation noises are known white Gaussian and the distribution of the initial state is also Gaussian (with unknown mean and covariance). For the first scenario, we formulate the problem as a risk minimization problem and show that its solution is statistically consistent. For the second scenario, we fit the problem into the framework of maximum-likelihood and Expectation Maximization (EM) algorithm is used to solve this problem. The performance for the estimations are shown by numerical examples.