Concept of both real and complex Gaussian Random Variables (GRVs) and their corresponding Gaussian Random processes (GRPs) is very important and critical for understanding and designing of a real Communication System. In this paper, we thus discuss and compute the various parameters involved in characterizing the real and complex Gaussian distributions completely for one and multiple Random Variables (RVs). We first carry out our analysis for one real GRV and then extend our work to multiple real GRVs. Furthermore, we apply our technique for studying one complex GRV and subsequently extend it to the analysis of multiple complex GRVs also known as multivariate complex Gaussian random vector. Finally, we make the necessary comparison between real and complex Gaussian Distributions which are the Key components of Additive White Gaussian Noise (AWGN) for baseband and bandpass transmission of the digital data through non ideal channels of digital Communication Systems.