Gaussian Process Regression (GPR) is a powerful non-parametric method. However, GPR may perform poorly if the data are contaminated by outliers. To address the issue, we replace the Gaussian process with a Student-t process and introduce dependent Student-t noise in this paper, leading to a Student-t Process Regression with Dependent Student-t noise model (TPRD). Closed form expressions for the marginal likelihood and predictive distribution of TPRD are derived. Besides, TPRD gives a probabilistic interpretation to the Student-t Process Regression with the noise incorporated into its Kernel (TPRK), which is a common approach for the Student-t process regression. Moreover, we analyze the influence of different kernels. If the kernel meets a condition, called β-property here, the maximum marginal likelihood estimation of TPRD's hyperparameters is independent of the degrees of freedom ν of the Student-t process, which implies that GPR, TPRD and TPRK have exactly the same predictive mean. Empirically, the degrees of freedom ν could be regarded as a convergence accelerator, indicating that TPRD with a suitable ν performs faster than GPR. If the kernel does not have the β-property, TPRD has better performances than GPR, without additional computational cost. On benchmark datasets, the proposed results are verified.