In this work, we apply the concept about operator connection to consider bivariate random tensor means. We first extend classical Markov and Chebyshev inequalities from a random variable to a random tensor by establishing Markov inequality for tensors and Chebyshev inequality for tensors. These inequalities are applied to establish tail bounds for bivariate random tensor means represented by operator perspectives based on various types of connection functions: tensor increasing functions, tensor decreasing functions, and tensor concavity functions. We also consider tail bounds relations for the summation and product of eigenvalues based on majorization ordering of eigenvalues of bivariate random tensor means. This is Part I of our work about random tensor inequalities and tail bounds for bivariate random tensor mean. In our Part II, we will consider bivariate random tensor mean with respect to non-invertible random tensors and their applications.