This paper analyzes the statistical properties of the signal-to-noise ratio (SNR) at the output of the Capon's minimum variance distortionless response (MVDR) beamformers when operating over impulsive noises. Particularly, we consider the supervised case in which the receiver employs the regularized Tyler estimator in order to estimate the covariance matrix of the interference-plus-noise process using $n$ observations of size $N\times 1$. The choice for the regularized Tylor estimator (RTE) is motivated by its resilience to the presence of outliers and its regularization parameter that guarantees a good conditioning of the covariance estimate. Of particular interest in this paper is the derivation of the second order statistics of the SINR. To achieve this goal, we consider two different approaches. The first one is based on considering the classical regime, referred to as the $n$-large regime, in which $N$ is assumed to be fixed while $n$ grows to infinity. The second approach is built upon recent results developed within the framework of random matrix theory and assumes that $N$ and $n$ grow large together. Numerical results are provided in order to compare between the accuracies of each regime under different settings.